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Secondary 14M15, 14E30}\n \n\\title[Fano 3-folds from homogeneous vector bundles over Grassmannians]{Fano 3-folds from homogeneous vector bundles over Grassmannians}\n\n\\begin{document}\n\\begin{abstract}\nWe rework the Mori--Mukai classification of Fano 3-folds, by describing each of the 105 families via biregular models as zero loci of general global sections of homogeneous vector bundles over products of Grassmannians.\n\\end{abstract}\n\\maketitle\n\\section{Introduction}\n\n\\thispagestyle{empty}\n\nThe classification of Fano 3-folds is one of the most influential results in birational geometry. Out of the 105 families, 17 have Picard rank $\\rho=1$. They are usually called \\emph{prime}. Their classification was completed first by Iskovskikh \\cite{isk}, using the birational technique of the \\emph{double projection from a line}. The classification was reworked by Mukai \\cite{mukai}, using the biregular \\emph{vector bundle method}. Mukai was able to describe most of the prime Fano varieties as complete intersections in certain homogeneous or quasi-homogeneous varieties. The latter in turn can be embedded in Grassmannians as zero loci of sections of homogeneous vector bundles.\n\nMori and Mukai \\cite{morimukai} classified as well the 88 remaining families of Fano 3-folds with Picard rank $\\rho \\geq 2$. However, the proof has little in common with the vector bundle strategy, relying on the powerful birational Mori's theory of extremal rays.\n\nOne of the aims of this paper is to rewrite the entire classification of 3-folds in a biregular fashion, finding models for the non-prime Fano 3-folds which are akin to the Mukai's vector bundle ones.\nIn particular, for each of the 105 Fano $X$ we will look for a suitable embedding $X \\subset \\prod \\Gr(k_i, n_i)$, such that $X$ can be described as the zero locus of a general global section of a homogeneous vector bundle $\\mathcal{F}$ over $\\prod \\Gr(k_i, n_i)$.\n\nIn \\cite{corti} Coates, Corti, Galkin, and Kasprzyk carried out a similar program. In particular, they were able to write down each of the 105 Fano 3-folds as zero loci of sections of vector bundles over GIT quotients. In some cases, their \\emph{key varieties} are products of Grassmannians, and we decided to adopt their models. However, in many cases, their model of choice is a complete intersection in a toric variety, which was particularly suitable for their purpose of computing the quantum periods, with the aim of using ideas from mirror symmetry for further classification results.\n\nOur motivating purpose is instead to attack the classification of Fano varieties in higher dimension from a representation-theoretical angle. In \\cite{kuchle} K\\\"uchle classified Fano 4-folds of index 1 that can be obtained from completely reducible, homogeneous vector bundles over a single Grassmannian $\\Gr(k,n)$. The resulting 20 families are therefore a sort of higher dimensional analogue of the Mukai models for 3-folds obtained via the vector bundle method. One of the main advantages of K\\\"uchle's method is that it relies only on very simple combinatorial data as input, such as the weight of the representation corresponding to the bundle involved. Moreover, this description allows an efficient computation of the invariants of the Fano, such as the Hodge numbers, for example using in combination the Koszul complex and Borel--Weil--Bott Theorem on the ambient variety. Such methods can be easily automatised via computer algebra, and extended to the case of products of Grassmannians $\\prod \\Gr(k_i, n_i)$, to say the least. This is exactly what we did. This paper originated from the construction of 3-folds via these methods; in a series of subsequent projects, we plan to work on more classification-type results, in dimension 4 and above.\n\nAs an initial benchmark for our strategy, we wanted to check how many of the 105 3-folds could be described using our methods. We found out that all 105 of them are. Although we do not believe that the same will be true in dimension 4 and higher, we hope to be able to find out many new and interesting examples of non-prime Fano 4-folds.\n\n\n\\subsection*{Main results}\n\nThe results of the paper are partially summarised in the following theorem. In what follows and throughout the whole paper, the notation $\\mathscr{Z}(\\mathcal{F}) \\subset G$ will denote the zero locus of a general global section of the vector bundle $\\mathcal{F}$ in the variety $G$.\n\\begin{thm}\\label{mainthm}\nLet $X$ be a smooth Fano 3-fold. Then there exist an ambient variety $G=\\prod \\Gr(k_i,n_i)$, product of (possibly weighted) Grassmannians, and a homogeneous vector bundle $\\mathcal{F}$ on $G$ such that $X= \\mathscr{Z}(\\mathcal{F}) \\subset G.$\n\\end{thm}\nThe only Fano varieties requiring weighted Grassmannians (actually, a unique weighted projective space) in their description without any alternative description are 1--11, 2--1, and 10--1, the others involving only classical Grassmannians. The weighted projective space in question is $\\mathbb{P}(1^3,2,3)$. The Fano 1--11 is a section of $\\mathcal{O}(6)$ on the latter, 2--1 is a blow up of 1--11 and 10--1 is a linear section (multiplied with a $\\mathbb{P}^1$). Notice that for the Fano 1--11 (which was present in this form in Mukai's classification as well), $-K_X$ is not very ample. A few other weighted projective spaces appear, but for all of them we provide alternative descriptions.\n\nIn the statement of Theorem \\ref{mainthm} we have not specified any hypothesis on the vector bundle $\\mathcal{F}$. Our \\emph{gold standard} for a homogeneous vector bundle $\\mathcal{F}$ is to be completely reducible and globally generated. Bundles with these properties are particularly suitable when facing classification problems. \nFor 85 out of the 105 families, we managed to find a vector bundle of this form; for the remaining ones, we used homogeneous bundles which are extensions of some other homogeneous completely reducible ones, so that the description is slightly more complicated but still well within our range of techniques. Out of these 20 families, for 5 of them the vector bundle is particular: it\nis of the form $\\mathcal{F}= \\mathcal{F}' \\oplus \\mathcal{G}$ where $\\mathcal{G}$ is a line bundle with no global sections on the total space, but with sections on $\\mathscr{Z}(\\mathcal{F}')$. This happens when we need to blow up along a subvariety involving an exceptional divisor coming from a previous blow up. We deal with this phenomenon in Caveat \\ref{caveatBundle}.\n\nWe partially collect these refinements in the following theorem.\n\\begin{thm} \\label{thm:refined} Let $X$ be a Fano as in Theorem \\ref{mainthm}. Then\n\\begin{itemize}\n \\item For 102\/105 families of Fano there exists a description without weighted factors in $G$.\n \\item For 85\/105 families of Fano there exists a description such that the bundle $\\mathcal{F}$ is completely reducible.\n\\end{itemize}\n\\end{thm}\n\nThe two theorems are proven in Section \\ref{Fano3folds}, which we devote to the construction of the aforementioned families, except for those which are already known in the literature. We collect all the models in Section \\ref{tables}; we include models for Del Pezzo surfaces as well. All models are general in moduli.\n\nWe draw the reader's attention to Section \\ref{identifications} as well. This is mainly a collection of technical lemmas and results, and we believe that most of them are well-known to experts. \nNonetheless, some of them are of independent interest, as they provide a dictionary between zero loci of sections of vector bundles and birational geometry. They were quite useful for translating Mori--Mukai models into our descriptions, and we believe that they can and will be useful for higher dimensional analyses. In this line of thought, we also present a few results involving flag varieties, even if they play only a small role in what follows.\n\n\\subsection*{Our models}\n\nMori--Mukai characterisation of the 88 non-prime 3-folds often involves intricate birational descriptions. The typical situation consists in blowing up a simpler 3-fold along a curve. Whenever the curve is a complete intersection in the base 3-fold, finding a suitable model in a product of Grassmannians is almost algorithmic; when the curve is not, then we perform a delicate analysis to understand how the curve can be cut in the ambient Fano. Subsequently, Lemma \\ref{lem:blowup}, Corollary \\ref{cor:cayleycrit}, and Lemma \\ref{lem:blowDegeneracyLocus} allow us to describe the resulting 3-fold as a complete intersection in a suitable projective bundle. We then need to describe the latter as a zero locus of some vector bundle over a product of Grassmannians. In many cases, this is a straightforward procedure and the proof takes few lines. However, some projective bundles turn out to be particularly tricky, and we have to deal with them case-by-case. \n\nFor other Fano we need to blow up a variety along a subvariety of codimension at least 3. To handle these cases, we collect and develop a few results which allow us to characterise these blow ups in term of zero loci of sections.\n\nWe want to give here an introductory example of a Fano 3-fold whose description is not immediate, yet admits a quite simple description in our model. We compare the original Mori--Mukai approach and the Coates--Corti--Galkin--Kasprzyk one with ours.\n\nLet us consider the Fano of rank 2, number 16 in the Mori--Mukai list. Following the notation which will be adopted in our paper, we will call it 2--16.\n\n\\begin{description}[leftmargin=0pt]\n\\item[2--16, Mori--Mukai] Blow up of the complete intersection of two quadrics in $\\mathbb{P}^5$ in a conic $C$. Notice that $C$ is not a complete intersection in the ambient variety $\\mathbb{Q}_1 \\cap \\mathbb{Q}_2 \\subset \\mathbb{P}^5$.\n\\item[2--16, Coates--Corti--Galkin--Kasprzyk] A codimension-2 complete intersection\n$\\mathscr{Z}(L+M, 2M) \\subset F$ where $F$ has weight data\n\\[\n\\begin{array}{rrrrrrrl} \n\t\\multicolumn{1}{c}{s_0} & \n\t\\multicolumn{1}{c}{s_1} & \n\t\\multicolumn{1}{c}{s_2} & \n\t\\multicolumn{1}{c}{x} & \n\t\\multicolumn{1}{c}{x_3} & \n\t\\multicolumn{1}{c}{x_4} & \n\t\\multicolumn{1}{c}{x_5} & \\\\ \n\t\\cmidrule{1-7}\n\t1 & 1 & 1 & -1 & 0 & 0 & 0& \\hspace{1.5ex} L\\\\ \n\t0 & 0 & 0 & 1 & 1 & 1 & 1 & \\hspace{1.5ex} M \\\\\n\\end{array}\n\\]\n\\end{description}\n\nFinally, our description realises this Fano as the zero locus of a general section of a globally generated homogeneous vector bundle over a (non-toric) product of Grassmannians.\n\\begin{description}[leftmargin=0pt]\n\\item[2--16, our description] The zero locus \\[\\mathscr{Z}(\\mathcal{U}^{\\vee}_{\\Gr(2,4)}(1,0) \\oplus \\mathcal{O}(0,2)) \\subset \\mathbb{P}^2 \\times \\Gr(2,4),\\] where $\\mathcal{U}$ is the rank 2 tautological subbundle.\n\\end{description}\n\nOur construction methods often allow for multiple models. For instance, the above Fano 2--16 can be realised as well as \\[ \\mathscr{Z}(\\mathcal{O}(1,0) \\oplus \\mathcal{O}(0,2)) \\subset \\Fl(1,2,4). \\]\nIn order to preserve the compactness of this paper we usually decided to present only one model for each Fano variety, with notable exceptions whenever we found an alternative description too elegant not to include it, or whenever they were important intermediate steps in the identification of the model. Our choice of model depends on our personal taste. The criterion for an $X= \\mathscr{Z}(\\mathcal{F}) \\subset \\prod \\Gr(k_i, n_i)$ was to pick the model with either the smallest number of factors or with the rank of $\\mathcal{F}$ as low as possible. To mention an example in lower dimension, the Del Pezzo surface of degree 5 can be equivalently described as $\\mathscr{Z}(\\mathcal{O}(1,0,0,0,1)\\oplus \\mathcal{O}(0,1,0,0,1) \\oplus \\mathcal{O}(0,0,1,0,1) \\oplus \\mathcal{O}(0,0,0,1,1)) \\subset (\\mathbb{P}^1)^4 \\times \\mathbb{P}^2 $ or as $\\mathscr{Z}(\\mathcal{O}(1)^{\\oplus 4}) \\subset \\Gr(2,5)$. We will prefer the latter description to the former. \n\n\n\n\\subsection*{Further directions}\n\\label{futureDirections}\n\nAs mentioned in the first part of the introduction, our methods are built with the explicit intention of being applied in higher dimension. Over a single Grassmannian $\\Gr(k,n)$ homogeneous, completely reducible vector bundles can be written as direct sums of $\\Sigma_{\\alpha} \\mathcal{Q} \\otimes \\Sigma_{\\beta}\\mathcal{U}$, where $\\Sigma_{\\alpha}$ (resp.\\ $\\Sigma_{\\beta}$) denotes the Schur functor indexed by the non-increasing sequence $\\alpha$ (resp.\\ $\\beta$); a similar expression holds for flag varieties and their products. This makes to some extent possible a methodical search for varieties which are zero loci of sections of bundles of this form.\n\nWhat we plan to do in a series of subsequent works is to classify all Fano in dimension 4 that can be obtained in this way, comparing our results with the already existing known classes of Fano 4-folds (\\cite{batyrev, coates, kalashnikov}, to cite a few). We are confident that many new and interesting examples can be found in this way, and the results of this paper are for sure strong motivations. We are particularly interested in the case of 4-folds of index 1 with Picard rank as high as possible and which are not a product. The \\emph{champion} at the moment is the Fano of Picard rank 9 constructed by Casagrande, Codogni, and Fanelli in \\cite{casagrande}; see, e.g., \\cite{CasagrandeSurvey} for a survey of results on the topic.\n\nAnother case of interest are Fano varieties in higher dimension with special Hodge-theoretical properties. In particular, Fano varieties in any dimension of K3 type (in the sense of \\cite{eg2}) have recently been studied due to their possible links with hyperk\\\"ahler manifolds. Finally, we remark that zero loci are particular cases of degeneracy loci of morphisms between vector bundles. It is certainly possible to further extend the above program to this framework, which has already been explored from many points of view (see, e.g., \\cite{TanturriHilbert}), or even to the so-called orbital degeneracy loci, a recently introduced wider class of varieties \\cite{BFMT2,BFMT}.\n\n\n\n\\subsection*{Plan of the paper} \nSection \\ref{identifications} is where we establish our toolbox, and state or prove several lemmas, useful to translate the Mori--Mukai birational language into our biregular one, and vice versa. Section \\ref{computeinvariants} is devoted to explaining how we are able to compute the invariants for all the models we present. Section \\ref{Fano3folds} is the core of the paper. A detailed description of all the families which are not provided in the literature is given. Section \\ref{tables} contains the tables and recap all the results in a schematic and handy fashion.\n\n\n\\subsection*{Notation and conventions}Throughout the whole paper, the notation $\\mathscr{Z}(\\mathcal{F}) \\subset X$ denotes the zero locus of a general global section of the vector bundle $\\mathcal{F}$ in the variety $X$. We will denote by $X_d$ a general hypersurface of degree $d$ inside $X$.\n\nIf $E$ is a rank $r$ vector bundle over a variety $X$, we denote by $\\mathbb{P}_X(E)$ (or simply by $\\mathbb{P}(E)$ when no confusion can arise) the projective bundle $\\pi:\\mathrm{Proj}(\\Sym E^{\\vee}) \\rightarrow X$; we remark that we adopt the subspace notation, as in \\cite[Chapter 9]{EisenbudHarris3264}. If we denote by $\\mathcal{O}_{\\mathbb{P}(E)}(1)$ (or simply $\\mathcal{O}(1)$) the relatively ample line bundle, this yields $H^0(\\mathbb{P}(E),\\mathcal{O}_{\\mathbb{P}(E)}(1)) \\cong H^0(X, E^{\\vee})$. Moreover $\n\\omega_{\\mathbb{P}(E)} \\cong \\mathcal{O}_{\\mathbb{P}(E)}(-r) \\otimes \\pi^*(\\omega_X \\otimes \\det(E^{\\vee})\n$\nand, for any line bundle $L$, the isomorphism $\\mathbb{P}(E)\\cong \\mathbb{P}(E\\otimes L)$ induces $\\mathcal{O}_{\\mathbb{P} (E)}(1) \\otimes L^\\vee = \\mathcal{O}_{\\mathbb{P}(E\\otimes L)}(1)$.\n\nFor products of varieties $X_1 \\times X_2$, the expression $\\mathcal{F}_1 \\boxtimes \\mathcal{F}_2$ will denote the tensor product between the pullbacks of $\\mathcal{F}_i$ via the natural projections. For products of Grassmannians $\\Gr(k_1,n_1) \\times \\Gr(k_2,n_2)$, we will almost always adopt the short form $\\mathcal{O}(a,b):=\\mathcal{O}(a) \\boxtimes \\mathcal{O}(b)$; we will often omit the pullbacks when no confusion can arise, so that, e.g., $\\mathcal{Q}_{\\Gr(k_1,n_1)}(1,2)=\\mathcal{Q}_{\\Gr(k_1,n_1)}(1) \\boxtimes \\mathcal{O}_{\\Gr(k_2,n_2)}(2)$. \n\nBy $\\Fl(k_1, \\ldots, k_r, n)$ we will denote the flag variety of subspaces $V_{k_1} \\subset V_{k_2} \\subset \\ldots \\subset V_{k_r} \\subset \\mathbb{C}^n$. We will denote by $\\pi_i$ the projection to the $i$-th Grassmannian $\\Gr(k_i,n)$. $\\mathcal{U}_i$ and $\\mathcal{Q}_i$ will denote the pullback of the tautological bundles via $\\pi_i$. For short, we will write $\\mathcal{O}(a,b)=\\pi_1^*(\\mathcal{O}(a)) \\otimes \\pi_2^*(\\mathcal{O}(b))$. In the rare cases where a flag is involved in a product of varieties, the different Picard groups will be separated by a semicolon, i.e., $\\mathcal{O}(a,b;c)=\\mathcal{O}(a,b) \\boxtimes \\mathcal{O}(c)$ on $\\Fl(k_1,k_2,n) \\times \\Gr(k',n')$.\n\nMany data for Table \\ref{tab:3folds} (and for the paper overall) are taken from \\cite{fanography}. They rely on the tables from \\cite{isp5,pcs,kps,cfst}. Many other alternative descriptions are taken from \\cite{corti}. We include the relevant citation to the alternative description in the table whenever appropriate. The notation X--Y for a Fano means a Fano of Picard rank $X$ which is the number $Y$ in the Mori--Mukai list.\nFinally, $\\mathbb{Q}_3$ denotes the 3-dimensional quadric hypersurface (Fano 1--16) and $\\mathbb{V}_5$ denotes the index 2, degree 5 linear section of $\\Gr(2,5)$ (Fano 1--15).\n\\subsection*{Acknowledgements} \nWe are indebted to Daniele Faenzi for many enlightening suggestions. Thanks to Vladimiro Benedetti, Marcello Bernardara, Giovanni Mongardi, and Miles Reid for useful discussions. EF and FT were partially supported by a ``Research in Paris'' grant held at Institut Henri Poincar\\'e. We thank the institute for the excellent working conditions. We acknowledge the Laboratoire Paul Painlev\\'e -- Universit\\'e de Lille, the Dipartimento di Matematica ``Giuseppe Peano'' -- Universit\\`a di Torino and INdAM for partial support as well. All three authors are members of INdAM-GNSAGA.\n\n\\section{Identifications}\n\\label{identifications}\n\nMost of the Fano 3-folds with Picard rank $\\rho\\geq 2$ arise as blow up of other Fano 3-folds with centre in distinguished subvarieties. Sometimes other standard birational descriptions are involved. The purpose of this subsection is therefore to establish a toolbox that allow us to translate the Mori--Mukai birational language into models suitable for our type of descriptions. Most of the lemmas appearing in this section are probably well-known to the experts: however for some of them we have not been able to locate clear proofs in the literature. \n\nThe most basic result is the description of the blow up of a projective space in a linear subspace. We will use the following lemma:\n\n\n\\begin{lemma} \\label{lem:blow}\nLet $\\mathcal{Q}$ be the tautological quotient bundle on $ \\mathbb{P}^{n-r}$. We have\n\\[\n\\Bl_{\\mathbb{P}^{r-1}}\\mathbb{P}^n \\cong \\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^{n-r}}(0,1)) \\subset \\mathbb{P}^{n-r} \\times \\mathbb{P}^n.\n\\]\n\n\\begin{proof}\nLet $V$ be a $n+1$-dimensional vector space such that $\\mathbb{P}^n \\cong \\mathbb{P}(V)$. By \\cite[Proposition 9.11]{EisenbudHarris3264},\n$\\Bl_{\\mathbb{P}^{r-1}}\\mathbb{P}^n$ is isomorphic to the projectivization of the vector bundle\n\\[\nE = \\mathcal{O}_{\\mathbb{P}^{n-r}}(-1) \\oplus (V' \\otimes \\mathcal{O}_{\\mathbb{P}^{n-r}}),\n\\]\nwhere $V' \\subset V$ has dimension $r$ and $\\mathbb{P}^{n-r}$ is identified with $\\mathbb{P}(V\/V')$. The bundle $E$ fits into the short exact sequence\n\\[\n0 \\rightarrow E \\rightarrow (V\/V' \\oplus V') \\otimes \\mathcal{O}_{\\mathbb{P}^{n-r}} \\rightarrow \\mathcal{Q}_{\\mathbb{P}^{n-r}} \\rightarrow 0,\n\\]\nhence $\\mathbb{P}(E)$ can be also expressed as the zero locus of $\\mathcal{Q} \\boxtimes \\mathcal{O}(1)$ inside $\\mathbb{P}^{n-r} \\times (\\mathbb{P}^n\\cong \\mathbb{P}(V\/V' \\oplus V'))$, as claimed.\n\\end{proof}\n\\end{lemma}\n\nIn the above lemma we used the fact that, as soon as we have a short exact sequence on $X$ of vector bundles\n$0 \\rightarrow\nE \\rightarrow\nF \\rightarrow\nG \\rightarrow\n0,$\nthen $\\mathbb{P}(E)$ can be obtained as the zero locus of a section of $t$ of $\\pi^*(G) \\otimes \\mathcal{O}_{\\mathbb{P}(F)}(1)$ over $\\pi:\\mathbb{P}(F)\\rightarrow X$. If $H^1(E)=0$, then $t$ can be chosen to be general; a particularly favourable situation will occur when $F\\cong \\mathcal{O}^{\\oplus r}$, so that $\\mathbb{P}(E)$ embeds into $X \\times \\mathbb{P}^{r-1}$.\n\nLemma \\ref{lem:blow} can be generalised for the Grassmannians context.\n\\begin{lemma}\\label{lem:blowInGrass}\nWe have\n\\[\n\\Bl_{\\Gr(k-1, n-1)}\\Gr(k,n) \\cong \\mathscr{Z}(\\mathcal{Q} \\boxtimes \\mathcal{U}^{\\vee}) \\subset \\Gr(k,n-1) \\times \\Gr(k,n),\n\\]\nwhere the centre of the blow up $\\Gr(k-1, n-1)$ is identified with $\\mathscr{Z}(\\mathcal{Q})\\subset \\Gr(k, n)$.\n\\begin{proof}\nLet $V_n, V_{n-1}$ be complex vector spaces of dimension $n, n-1$ respectively. A section of $\\mathcal{Q} \\boxtimes \\mathcal{U}^{\\vee}$ over $\\Gr(k,V_{n-1}) \\times \\Gr(k,V_n)$ can be regarded as a section of $\\mathcal{U}^{\\vee} \\boxtimes \\mathcal{U}^{\\vee}$ over $\\Gr(n-k-1,V_{n-1}^\\vee) \\times \\Gr(k,V_n)$, and the corresponding zero loci are canonically isomorphic.\n\nA section of the latter vector bundle is of the form\n\\[\ns=f_1 x_1 + \\ldots + f_{n-1} x_{n-1}\n\\]\nfor some $f_i \\in V_{n-1}$, $x_i \\in V_n^\\vee$. Let us fix bases for $V_{n-1}^{\\vee}, V_n$ accordingly. Up to the action of $\\GL_k$, a point in $\\Gr(k,V_n)$ is represented by a $k \\times n$ matrix\n\\[\nA=\\left( \\begin{array}{ccc}\na_{1,1} & \\cdots & a_{1,n}\\\\\n\\vdots & & \\vdots \\\\\na_{k,1} & \\cdots & a_{k,n}\n\\end{array}\n\\right)\n\\]\nand a section $x_i \\in V_n^\\vee$ evaluates as $x_i (A)=(a_{1,i} \\cdots a_{k,i})^t$. Analogously, a section $f_i \\in V_n$ evaluates on a point $B \\in \\Gr(n-k-1,V_{n-1}^\\vee)$, seen as a $n-k-1 \\times n-1$ matrix, as $f_i(B)=(b_{1,i} \\cdots b_{n-k-1,i})^t$.\n\nThe evaluation of a section $f_i x_i$ on a point $(A,B)$ is given by the $k \\times n-k-1$ matrix $x_i(A) \\cdot (f_i(B))^t$. It is straightforward to check that\n\\[\ns(A,B) = 0 \\qquad \\mbox{if and only if} \\qquad A \\cdot \\left( \n\\begin{array}{c}\nB^t \\\\\n\\hline 0 \\cdots 0\n\\end{array} \\right)=0.\n\\]\n\nLet $Y$ be the zero locus of $s$. We want to study the fibres of the (restriction of the) natural projection $Y \\rightarrow \\Gr(k,V_n)$. This amounts to solving a linear system with $b_{1,1}, b_{1,2}, \\ldots, b_{1,n-1}, b_{2,1}, \\ldots, b_{n-k-1,n-1}$ as variables. With this choice of coordinates, the matrix associated to the linear system is the $(n-1)(n-k-1) \\times k(n-k-1)$ matrix\n\\[\n\\left(\n\\begin{array}{cccc}\n\\tilde{A} \\\\\n& \\tilde{A} \\\\\n& & \\ddots \\\\\n& & & \\tilde{A}\n\\end{array}\n\\right), \\qquad \\mbox{where } A=\\left(\n\\begin{array}{c|c}\n\\tilde{A} & \\begin{array}{c}\na_{1,n}\\\\\n\\vdots\\\\\na_{k,n}\n\\end{array}\n\\end{array}\n\\right).\n\\]\nThe fiber over a general point $A$, i.e., whenever $\\tilde{A}$ has maximal rank, is a single point $\\in \\Gr(n-k-1,V_{n-1}^\\vee)$, hence $Y \\rightarrow \\Gr(k,V_n)$ is birational. The fiber over $A$ is positive-dimensional if and only if $\\tilde{A}$ has rank at most $k-1$, i.e., if and only if\n\\[\n\\rank \\left(\n\\begin{array}{c}\nA \\\\\n\\hline 0 \\cdots 0 \\, 1\n\\end{array}\n\\right)0$, which implies that $H^0(\\Fl, E)$ surjects onto $H^0(Y, E|_Y)$.\n\n\nWe do a recap in the following handy corollary. This is will be useful to deal with the case of complete intersection curves of different degrees.\n\\begin{corollary}\\label{cor:cayleycrit}\nAssume that we have a bundle $F=E \\oplus G$ on $\\Fl(1,2,n)$, with $G=\\pi_2^*\\widetilde{G}$ for a bundle $\\widetilde{G}$ on $\\Gr(2,n)$ and $X=\\mathscr{Z}(F) \\subset Y= \\mathscr{Z}(G) \\subset \\Fl(1,2,n)$. Denote by $\\widetilde{Y}$ the zero locus $\\widetilde{Y}=\\mathscr{Z}(\\widetilde{G}) \\subset \\Gr(2,n)$. Assume that\n$H^0(Y, E|_Y) \\cong H^0(\\widetilde{Y},\\mathcal{U}|_{\\widetilde{Y}} \\otimes L)$ for some line bundle $L$ on $\\widetilde{Y}$. Denote by $\\widetilde{X}=\\mathscr{Z}(\\mathcal{U}|_{\\widetilde{Y}} \\otimes L)\\subset \\widetilde{Y}$. Then $X \\cong \\Bl_{\\widetilde{X}} \\widetilde{Y}$.\n\\end{corollary}\n\nThere is a further generalisation of the Cayley trick, that applies to degeneracy loci as well, which we recall for completeness.\n\n\\begin{lemma}[{\\cite[Lemma 2.1]{kuznetsovKuchle}}]\n\\label{lem:blowDegeneracyLocus}\nLet $\\varphi:E \\rightarrow F$ be a morphism of vector bundles of ranks $r+1$, $r$ respectively on a Cohen--Macaulay variety $X$.\nDenote by $D_k(\\varphi)$ the $k$-th degeneracy locus of $\\varphi$, i.e., the locus where the morphism has corank at least $k$.\nConsider the projectivization $\\pi:\\mathbb{P}(E) \\rightarrow X$, then $\\varphi$ gives a global section of the vector bundle $\\pi^*F \\otimes \\mathcal{O}(1)$.\nIf $\\codim D_k(\\varphi) \\ge k + 1$ for all $k \\ge 1$ then the zero locus of $\\varphi$ on $\\mathbb{P}(E)$ is isomorphic to the blow up of $X$ along $D_1(\\varphi)$.\n\\end{lemma}\n\nIn practice, we will often need to find some projective bundle $\\mathbb{P}(\\mathcal{O}(-d_1,\\dotsc,-d_m) \\oplus \\mathcal{O}^{\\oplus r})$ as the zero locus of a suitable vector bundle over a product of Grassmannians. The following remark will be very helpful for this sake; an instance of its application will be Lemma \\ref{projBundle1-12}. \n\n\\begin{rmk}\n\\label{rem:principalParts} Let $L$ be a line bundle on $X$. For any $k$ one can define $\\mathcal{P}^k(L)$, the bundle of $k$-principal parts of $L$, of rank $\\binom{k+\\ddim X}{k}$. One has $\\mathcal{P}^0(L)=L$; by \\cite[Exp II, Appendix II 1.2.4.]{SGA} there exist natural short exact sequences\n\\begin{equation}\\label{seq:principalparts} 0 \\rightarrow \\Sym^k (\\Omega_X)(L) \\rightarrow \\mathcal{P}^k(L) \\rightarrow \\mathcal{P}^{k-1}(L) \\rightarrow 0. \\end{equation}\nIf $X \\cong \\mathbb{P}^n$ these bundles of principal parts are homogeneous, and in \\cite[Thm 1.1]{re} their splitting type is determined. The situation is particularly simple when we consider $L=\\mathcal{O}_{\\mathbb{P}^n}(d)$ with $d \\geq k$: in this case one has $\\mathcal{P}^k(\\mathcal{O}_{\\mathbb{P}^n}(d)) \\cong \\Sym^k V_{n+1} \\otimes \\mathcal{O}_{\\mathbb{P}^n}(d-k)$. The sequence above for $k=1$ coincides with the dualised twisted Euler sequence.\n\\end{rmk}\n\n\n\n\nWe finish this section with a classical remark on how to characterise double covers as hypersurfaces in projective bundles. A detailed proof can be found for example in \\cite[Lemma 1.2]{lyu}. The formula below can be easily generalised to the case of $k$-cyclic covers, using $\\mathcal{O}_P(k)$ instead.\n\n\\begin{rmk}\\label{lem:doublecovers}\nLet $X$ be a 2-fold cyclic covering of $Y$, ramified along a smooth divisor $D$, and $L$ a line bundle with $L^{\\otimes 2}=\\mathcal{O}_Y(D)$, which is assumed to be $2$-divisible in $\\mathrm{Pic}(Y)$. Then $X$ can be identified with $\\mathscr{Z}(\\mathcal{O}_P(2))$ in $P:=\\mathbb{P}_Y(\\mathcal{O} \\oplus L^{\\vee})$. \n\\end{rmk}\n\n\n\\section{How to compute invariants}\n\\label{computeinvariants}\nIn this section we explain and show with a concrete example how we can compute the invariants of a zero locus of a general section of a given homogeneous vector bundle on a product of flag varieties.\n\nAs a matter of fact, such computations are not strictly necessary for the identification of the models we found for the Fano 3-folds in the next section. However, we want to stress out the importance of having such a tool for two reasons. On the one hand, one could start producing in an automatised way many examples coming from homogeneous vector bundles on products of flag varieties and later try to identify them using the existing classifications. This was exactly the starting point of this project and what made us able to characterise, along the process, many zero loci of sections from a geometric point of view. Several results of Section \\ref{identifications} have been found by trying to generalise the evidences coming from all the examples we had. On the other hand, it goes without saying that these methods will certainly be very useful when a similar search will be performed for varieties which have not yet been classified.\n\n\\subsection{The invariants \\texorpdfstring{$h^0(-K)$ and $(-K)^3$}{h\\^{}0(-K) and (-K)\\^{}3}}\n\nThese invariants can be computed via intersection theory. If $X$ is a product of flag varieties, then we know its graded intersection ring of algebraic cycle classes modulo numerical equivalence. We know how to integrate, so that Hirzebruch--Riemann--Roch Theorem yields a way to compute $\\chi(E)$ for any vector bundle $E$ with assigned Chern classes.\n\nThe situation does not change much when we consider a subvariety $\\mathscr{Z}(\\mathcal{F}) \\subset X$ given as the zero locus of a general section of some vector bundle $\\mathcal{F}$ on $X$. If we know the Chern classes of $\\mathcal{F}$, we can write down the graded intersection ring of $\\mathscr{Z}(\\mathcal{F})$, as well as count points on $0$-dimensional cycles.\n\nIn concrete examples, instead of doing computations by hand, it is of course convenient to use some computer algebra software. Our choice fell on \\cite{M2}, for which an already developed package \\cite{Schubert2} implementing the methods we need is available. This allows us to compute $(-K_{\\mathscr{Z}(\\mathcal{F})})^3$, as the Chern classes of the canonical sheaf of $\\mathscr{Z}(\\mathcal{F})$ are easy to express. As for $h^0(-K_{\\mathscr{Z}(\\mathcal{F})})$, we certainly know how to compute $\\chi(-K_{\\mathscr{Z}(\\mathcal{F})})$. But $-K=K-2K$ and $-2K$ is ample, so the Kodaira Vanishing Theorem implies $h^{i>0}(-K_{\\mathscr{Z}(\\mathcal{F})})=0$.\n\n\\subsection{Hodge numbers and tangent cohomology}\n\nBeside the aforementioned invariants, one of the most important data one would like to know about a Fano variety is its Picard rank. More in general, it is rather important to compute $h^{i,j}$ for a given variety. In our setting this is perfectly doable using classical tools as the Koszul complex and a bit of representation theory, even though the computations may quickly become cumbersome if the involved vector bundles have high rank or several summands.\n\nWe briefly recall the strategy. Let us suppose that $Y=\\mathscr{Z}(\\mathcal{F})\\subset X$. Assume that $\\rank(\\mathcal{F})=r$. For each $j \\in \\mathbb{N}$, we have the $j$-th exterior power of the conormal sequence\n\\begin{equation}\n\\label{wedgeKConormal}\n0\\rightarrow\n\\Sym^j \\mathcal{F}^\\vee|_Y \\rightarrow\n(\\Sym^{j-1} \\mathcal{F}^\\vee \\otimes \\Omega_X)|_Y \\rightarrow\n\\dotso \\rightarrow\n(\\Sym^{j-k} \\mathcal{F}^\\vee \\otimes \\Omega^k_X)|_Y \\rightarrow\n\\dotso \\rightarrow\n\\Omega^j_X|_Y \\rightarrow\n\\Omega^j_Y \\rightarrow\n0.\n\\end{equation}\nAs our goal is $h^i(\\Omega^j_Y)$, we can compute the dimensions of the cohomology groups of all the other terms in \\eqref{wedgeKConormal}, split it into short exact sequences and use the induced long exact sequences in cohomology to get the result.\n\nEach term $(\\Sym^{j-k} \\mathcal{F}^\\vee \\otimes \\Omega^k_X)|_Y$ is in turn resolved by an exact Koszul complex\n\\begin{equation*}\n0\\rightarrow\n\\bigwedge^r \\mathcal{F}^\\vee \\otimes \\Sym^{j-k} \\mathcal{F}^\\vee \\otimes \\Omega^k_X \\rightarrow\n\\dotso \\rightarrow\n\\mathcal{F}^\\vee \\otimes \\Sym^{j-k} \\mathcal{F}^\\vee \\otimes \\Omega^k_X \\rightarrow\n\\Sym^{j-k} \\mathcal{F}^\\vee \\otimes \\Omega^k_X,\n\\end{equation*}\nso that we are led to compute the cohomology groups of the terms above. If $X$ is a product of Grassmannians and $F$ is completely reducible, then those terms are completely reducible as well: a decomposition can be found via suitable plethysms. The cohomology groups can be then obtained via the usual Borel--Weil--Bott Theorem. \n\nThings get worse if $X$ has some genuine flag variety as a factor, in which case $\\Omega_X$ is an extension of completely reducible vector bundles, or if $F$ itself is an extension thereof. In these cases, one needs to deal carefully with the exterior\/symmetric power of an extension (which is an extension itself) and the tensor product of extensions; in the end, each term of the Koszul complex above is again an extension of completely reducible vector bundles, whose cohomology groups can be easily computed and arranged to get the result.\n\nIt may happen that several cohomology groups do not vanish, so that in the induced long exact sequences in cohomology there are boundary homomorphisms whose rank is a priori not known. This leads to some ambiguity in the final results, and can be partially solved by considering the additional relations involving $h^{i,j}$ such as the symmetries in the Hodge diamond and the computation of $\\chi(\\Omega^j_Y)$ as done above.\n\nAdditionally, suppose that we want to get some information on the automorphism group and the space of deformations of $Y$. One way is to compute $h^0(T_Y)$ and $h^1(T_Y)$ via the normal sequence\n\\[\n0 \\rightarrow\nT_Y \\rightarrow\nT_X|_Y \\rightarrow\n\\mathcal{F}|_Y \\rightarrow\n0.\n\\]\nAs before, one can compute the cohomology groups of the terms on the right via the usual Koszul complex and get some information on $h^i(T_Y)$.\n\nA rather easy example of application of the whole routine is provided in Section \\ref{aworkedexample}. It is evident that such computations cannot be done by hand for more complicated examples, especially for a significant number of cases. A Macaulay2 \\cite{M2} package which was developed to implement and automatise the procedure just described will be presented in \\cite{FatighentiTanturriPackage}.\n\n\\subsection{A worked example}\n\\label{aworkedexample}\n\nLet us show how to concretely compute the Hodge numbers of $Y:= \\mathscr{Z}(\\mathcal{U}^{\\vee}_{\\Gr(2,4)}(1,0) \\oplus \\mathcal{O}(0,2)) \\subset \\mathbb{P}^2 \\times \\Gr(2,4)=:X$, which we will prove to be a model for \\hyperlink{Fano2--16}{2--16}.\n\nOur vector bundle $\\mathcal{F}:=\\mathcal{U}^{\\vee}_{\\Gr(2,4)}(1,0) \\oplus \\mathcal{O}(0,2)$ has rank $3$. For $j=0$, \\eqref{wedgeKConormal} simply becomes $\\mathcal{O}_Y \\rightarrow \\mathcal{O}_Y$. The Koszul complex resolving $\\mathcal{O}_Y$ is\n\\[\n0 \\rightarrow\n\\mathcal{O}(-2,-3) \\rightarrow\n\\mathcal{U}_{\\Gr(2,4)}(-1,-2) \\oplus \\mathcal{O}(-2,-1) \\rightarrow\n\\mathcal{O}(0,-2) \\oplus \\mathcal{U}_{\\Gr(2,4)}(-1,0) \\rightarrow\n\\mathcal{O};\n\\]\nthe only non-zero cohomology group is $H^0(\\mathcal{O})\\cong \\mathbb{C}$, which gives $h^{0,0}=1$ and $h^{0,j}=0$ for $j>0$.\n\nFor $j=1$, \\eqref{wedgeKConormal} yields the usual conormal short exact sequence. The term on the left is $\\mathcal{F}^\\vee|_Y$, which is resolved by a Koszul complex whose terms are\n\\begin{equation*}\n\\begin{array}{rcl}\n\\bigwedge^3 \\mathcal{F}^\\vee \\otimes \\mathcal{F}^\\vee & = &\\mathcal{O}(-2,-5) \\oplus \\mathcal{U}_{\\Gr(2,4)}(-3,-3) \\\\\n\\bigwedge^2 \\mathcal{F}^\\vee \\otimes \\mathcal{F}^\\vee & = & \\mathcal{U}_{\\Gr(2,4)}(-1,-4) \\oplus \\Sym^2\\mathcal{U}_{\\Gr(2,4)}(-2,-2)\\oplus \\mathcal{O}(-2,-3)^{\\oplus 2} \\oplus\\mathcal{U}_{\\Gr(2,4)}(-3,-1) \\\\\n\\mathcal{F}^\\vee \\otimes \\mathcal{F}^\\vee & = & \\mathcal{O}(0,-4) \\oplus \n\\mathcal{U}_{\\Gr(2,4)}(-1,-2)^{\\oplus 2} \\oplus \n\\Sym^2 \\mathcal{U}_{\\Gr(2,4)}(-2,0) \\oplus \n\\mathcal{O}(-2,-1)\\\\\n\\mathcal{F}^\\vee & = & \\mathcal{O}(0,-2) \\oplus \n\\mathcal{U}_{\\Gr(2,4)}(-1,0);\n\\end{array}\n\\end{equation*}\nthe only non-zero cohomology group is $h^4(\\mathcal{F}^\\vee \\otimes \\mathcal{F}^\\vee)=1$, which yields $h^3(\\mathcal{F}^\\vee|_Y)=1$.\n\nThe middle term is $\\Omega_X|_Y$, which is resolved by a Koszul complex whose terms are\n\\begin{equation*}\n\\begin{array}{rcl}\n\\bigwedge^3 \\mathcal{F}^\\vee \\otimes \\Omega_X &= & \n\\mathcal{Q}_{\\Gr(2,4)} \\otimes \\mathcal{U}_{\\Gr(2,4)}(-2,-4) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(-4,-3)\n\\\\\n\\bigwedge^2 \\mathcal{F}^\\vee \\otimes \\Omega_X & = & \n\\mathcal{Q}_{\\Gr(2,4)}\\otimes\\left(\\Sym^2\\mathcal{U}_{\\Gr(2,4)}(-1,-3)\n\\oplus \\mathcal{O}(-1,-4) \\oplus \\mathcal{U}_{\\Gr(2,4)}(-2,-2)\n\\right) \\oplus \\\\ & & {}\\oplus\n \\mathcal{Q}_{\\mathbb{P}^2}(-4,-1) \\oplus \\mathcal{U}_{\\Gr(2,4)} \\otimes \\mathcal{Q}_{\\mathbb{P}^2}(-3,-2)\n\\\\\n\\mathcal{F}^\\vee \\otimes \\Omega_X & = & \n\\mathcal{Q}_{\\Gr(2,4)}\\otimes\\left(\\mathcal{U}_{\\Gr(2,4)}(0,-3)\n\\oplus \\Sym^2\\mathcal{U}_{\\Gr(2,4)}(-1,-1)\n\\oplus \\mathcal{O}(-1,-2)\n\\right) \\oplus \\\\ & & {}\\oplus\n \\mathcal{Q}_{\\mathbb{P}^2}(-2,-2) \\oplus \\mathcal{U}_{\\Gr(2,4)}\\otimes \\mathcal{Q}_{\\mathbb{P}^2}(-3,0)\\\\\n\\Omega_X & = & \\mathcal{Q}_{\\Gr(2,4)}\\otimes \\mathcal{U}_{\\Gr(2,4)}(0,-1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(-2,0);\n\\end{array}\n\\end{equation*}\nthe only non-zero cohomology groups are $h^3(\\mathcal{F}^\\vee \\otimes \\Omega_X)=1$ and $h^1(\\Omega_X)=2$, which yield $h^1(\\Omega_X|_Y)=2$ and $h^2(\\Omega_X|_Y)=1$.\n\nThe long exact sequence in cohomology induced by the conormal sequence then gives $h^{1,1}=2$ and $h^{1,2}=2$, while the other $h^{1,j}$ are zero.\n\n\nSimilar computations can be performed to compute $h^i(T_Y)$, by considering the normal sequence. The middle term is $T_X|_Y$, which is resolved by a Koszul complex whose terms are\n\\begin{equation*}\n\\begin{array}{rcl}\n\\bigwedge^3 \\mathcal{F}^\\vee \\otimes T_X &= & \n\\mathcal{Q}_{\\Gr(2,4)} \\otimes \\mathcal{U}_{\\Gr(2,4)}(-2,-2) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(-1,-3) \n\\\\\n\\bigwedge^2 \\mathcal{F}^\\vee \\otimes T_X & = & \n\\mathcal{Q}_{\\Gr(2,4)} \\otimes \\left( \n\\Sym^2 \\mathcal{U}_{\\Gr(2,4)}(-1,-1) \\oplus \\mathcal{O}(-1,-2) \\oplus \\mathcal{U}_{\\Gr(2,4)}(-2,0)\n\\right) \\oplus \\\\ & & {}\\oplus\n\\mathcal{Q}_{\\mathbb{P}^2} \\otimes \\left( \n\\mathcal{U}_{\\Gr(2,4)}(0,-2) \\oplus \\mathcal{O}(-1,-1) \n\\right)\n\\\\\n\\mathcal{F}^\\vee \\otimes T_X & = & \n\\mathcal{Q}_{\\Gr(2,4)} \\otimes \\left(\n\\mathcal{U}_{\\Gr(2,4)}(0,-1) \\oplus\n\\Sym^2 \\mathcal{U}_{\\Gr(2,4)}(-1,1) \\oplus\n\\mathcal{O}(-1,0)\n\\right) \\oplus \\\\ & & {}\\oplus\n\\mathcal{Q}_{\\mathbb{P}^2} \\otimes \\left( \n\\mathcal{O}(1,-2) \\oplus \\mathcal{U}_{\\Gr(2,4)}\n\\right)\n\\\\\nT_X & = & \n\\mathcal{Q}_{\\Gr(2,4)} \\otimes \\mathcal{U}_{\\Gr(2,4)}(0,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(1,0)\n;\n\\end{array}\n\\end{equation*}\nthe only non-zero cohomology groups are \n$h^1(\\mathcal{F}^\\vee \\otimes T_X)=1$ and $h^0(T_X)=23$, which yield $h^0(T_X|_Y)=24$. Similarly, the term on the right is $\\mathcal{F}|_Y$, which is resolved by a Koszul complex whose terms are\n\\begin{equation*}\n\\begin{array}{rcl}\n\\bigwedge^3 \\mathcal{F}^\\vee \\otimes \\mathcal{F} &= & \n\\mathcal{O}(-2,-1) \\oplus \\mathcal{U}_{\\Gr(2,4)}(-1,-2)\n\\\\\n\\bigwedge^2 \\mathcal{F}^\\vee \\otimes \\mathcal{F} & = & \n\\mathcal{U}_{\\Gr(2,4)}(-1,0) \\oplus \\mathcal{O}(-2,1) \\oplus \\Sym^2 \\mathcal{U}_{\\Gr(2,4)}(0,-1) \\oplus \\mathcal{O}(0,-2) \\oplus \\mathcal{U}_{\\Gr(2,4)}(-1,0)\n\\\\\n\\mathcal{F}^\\vee \\otimes \\mathcal{F} & = & \n\\mathcal{O}^{\\oplus 2} \\oplus \\mathcal{U}_{\\Gr(2,4)}(-1,2) \\oplus \\mathcal{U}_{\\Gr(2,4)}(1,-1) \\oplus \\Sym^2 \\mathcal{U}_{\\Gr(2,4)}(0,1)\n\\\\\n\\mathcal{F} & = & \n\\mathcal{O}(0,2) \\oplus \\mathcal{U}_{\\Gr(2,4)}(1,1)\n;\n\\end{array}\n\\end{equation*}\nthe only non-zero cohomology groups are \n$h^2(\\bigwedge^2 \\mathcal{F}^\\vee \\otimes \\mathcal{F})=1$, $h^0(\\mathcal{F}^\\vee \\otimes \\mathcal{F})=2$, and $h^0(\\mathcal{F})=32$, which yield $h^0(\\mathcal{F}|_Y)=31$.\n\nThus, $h^1(T_Y)-h^0(T_Y)=31-24=7$, and indeed Fano \\hyperlink{Fano2--16}{2--16} is known to have a $7$-dimensional moduli space.\n\n\n\\section{Fano 3-folds as zero loci of sections}\n\\label{Fano3folds}\n\nIn this section a model for each Fano 3-fold as the zero locus of a general section of a vector bundle over a product of Grassmannians is given, provided that such a description is not available in the literature. For each model we prove the identification with the corresponding Fano; all the examples have been checked to have the right Hodge diamond and invariants as described in Section \\ref{computeinvariants}.\n\n\n\\hypertarget{Fano1--1}{\\subsection*{Fano 1--1}}\n\\subsubsection*{Mori-Mukai} Double cover of $\\mathbb{P}^3$ with branch locus a divisor of degree 6.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(6)) \\subset \\mathbb{P}(1,1,1,1,3)$.\n\\subsubsection*{Identification} The obvious description as weighted hypersurface is classical. We want to give however another description embedded in a product of non-weighted Grassmannians.\n\nBy Lemma \\ref{lem:doublecovers}, we can express our Fano as the zero locus of $\\mathcal{O}(2)$ over $\\mathbb{P}_{\\mathbb{P}^3}(\\mathcal{O}(-3) \\oplus \\mathcal{O})$. We thus need to express such projective bundle as the zero locus of a section of a suitable vector bundle. To do that, we adopt a general strategy which will be explained in more details for \\hyperlink{Fano1--12}{1--12} or \\hyperlink{Fano2--2}{2--2}: we start from the short exact sequences provided by Remark \\ref{rem:principalParts}\n\\begin{equation}\n\\begin{gathered}\n0 \\rightarrow\n\\mathcal{O}(-3) \\rightarrow\n\\mathcal{O}(-2)^{\\oplus 4} \\rightarrow\n\\mathcal{Q}(-2) \\rightarrow\n0,\\\\\n0 \\rightarrow\n\\mathcal{O}(-2)^{\\oplus 4} \\rightarrow\n\\mathcal{O}(-1)^{\\oplus 10} \\rightarrow\n\\Sym^2\\mathcal{Q}(-1) \\rightarrow\n0,\\\\\n\\label{thirdseq}\n0 \\rightarrow\n\\mathcal{O}(-1)^{\\oplus 10} \\rightarrow\n\\mathcal{O}^{\\oplus 20} \\rightarrow\n\\Sym^3\\mathcal{Q} \\rightarrow\n0.\n\\end{gathered}\n\\end{equation}\nWe can arrange the first two using the snake lemma as in Lemma \\ref{projBundle1-12} or Lemma \\ref{projBundle2-2}: we get\n\\[\n0 \\rightarrow\n\\mathcal{O}(-3) \\rightarrow\n\\mathcal{O}(-1)^{\\oplus 10} \\rightarrow\n\\Lambda \\rightarrow\n0\n\\]\nfor a uniquely defined extension $\\Lambda \\in \\Ext^1(\\Sym^2\\mathcal{Q}(-1),\\mathcal{Q}(-2))$. The latter sequence can be again arranged with the third one in \\eqref{thirdseq}, to get\n\\[\n0 \\rightarrow\n\\mathcal{O}(-3) \\rightarrow\n\\mathcal{O}^{\\oplus 20} \\rightarrow\nK \\rightarrow\n0\n\\]\nfor another uniquely defined extension $K \\in \\Ext^1(\\Sym^3\\mathcal{Q},\\Lambda)$. Adding $\\mathcal{O} \\rightarrow \\mathcal{O}$ to the above sequence, we get that our Fano can be expressed as\n\\[\n\\mathscr{Z}(K(0,1) \\oplus \\mathcal{O}(0,2)) \\subset \\mathbb{P}^{3} \\times \\mathbb{P}^{20}.\n\\]\n\n\\hypertarget{Fano1--12}{\\subsection*{Fano 1--12}}\n\\subsubsection*{Mori-Mukai} Double cover of $\\mathbb{P}^3$ with branch locus a smooth quartic surface.\n\\subsubsection*{Our description} $\\mathscr{Z} (\\mathcal{O}(4)) \\subset \\mathbb{P}(1,1,1,1,2)$.\n\\subsubsection*{Identification}\nThe obvious description as weighted hypersurface is classical. We want to give however a rather simple description as a subvariety in a product of projective spaces. We notice that our Fano is, by Lemma \\ref{lem:doublecovers}, the zero locus of $\\mathcal{O}(2)$ on $\\mathbb{P}_{\\mathbb{P}^3}(\\mathcal{O}(-2) \\oplus \\mathcal{O})$.\n\n\\begin{lemma}\n\\label{projBundle1-12}\nThe projective bundle $\\mathbb{P}_{\\mathbb{P}^3}(\\mathcal{O}(-2) \\oplus \\mathcal{O})$ can be obtained as the zero locus of $\\Lambda(0,1)$ over $\\mathbb{P}^3 \\times \\mathbb{P}^{10}$, being $\\Lambda \\in \\Ext^1(\\Sym^2 \\mathcal{Q},\\mathcal{Q}(-1))$ a uniquely defined extension on $\\mathbb{P}^3$ fitting into sequence \\eqref{Lambda1--12} below.\n\\begin{proof}\nOur goal is to write $\\mathcal{O}_{\\mathbb{P}^3}(-2) \\oplus \\mathcal{O}_{\\mathbb{P}^3}$ as a subbundle of $\\mathcal{O}_{\\mathbb{P}^3}^{\\oplus 11}$. By Remark \\ref{rem:principalParts}, we have two (dual) canonical short exact sequences on $\\mathbb{P}^3$\n\\begin{gather*}\n0 \\rightarrow \\mathcal{O}(-2) \\rightarrow \\mathcal{O}(-1)^{\\oplus 4} \\rightarrow \\mathcal{Q}(-1) \\rightarrow 0,\\\\\n0 \\rightarrow \\mathcal{O}(-1)^{\\oplus 4} \\rightarrow \\mathcal{O}^{\\oplus 10} \\rightarrow \\Sym^2 \\mathcal{Q} \\rightarrow 0.\n\\end{gather*}\nThese fit as the first row and middle column of the exact diagram on $\\mathbb{P}^3$ here below, which can be completed by the snake lemma as\n\\begin{equation}\n\\label{snake1-12}\n\\begin{gathered}\n\\xymatrix{\n& 0 \\ar[d] & 0 \\ar[d] & 0 \\ar[d]\\\\\n0 \\ar[r] & \\mathcal{O}(-2) \\ar[d]^-= \\ar[r] & \\mathcal{O}(-1)^{\\oplus 4} \\ar[d] \\ar[r] & \\mathcal{Q}(-1) \\ar[d] \\ar[r] & 0 \\\\\n0 \\ar[r] & \\mathcal{O}(-2) \\ar[d] \\ar[r] & \\mathcal{O}^{\\oplus 10} \\ar[d] \\ar[r] & \\Lambda \\ar[d] \\ar[r] & 0 \\\\\n0 \\ar[r] & 0 \\ar[r]\\ar[d] & \\Sym^2 \\mathcal{Q} \\ar[r]^-= \\ar[d] & \\Sym^2 \\mathcal{Q} \\ar[r]\\ar[d] & 0 \\\\\n& 0 & 0 & 0 \\\\\n}\n\\end{gathered}\n\\end{equation}\nfor a uniquely determined homogeneous vector bundle $\\Lambda$ of rank $9$. The last column describes $\\Lambda$ as a non-split extension\n\\begin{equation}\n\\label{Lambda1--12}\n0 \\rightarrow \\mathcal{Q}_{\\mathbb{P}^3}(-1) \\rightarrow \\Lambda \\rightarrow \\Sym^2 \\mathcal{Q}_{\\mathbb{P}^3} \\rightarrow 0.\n\\end{equation}\n\nThe rank $9$ bundle $\\Lambda$ is homogeneous, not completely reducible and globally generated, and its space of global sections coincides with $H^0(\\mathbb{P}^3, \\Sym^2 \\mathcal{Q})\\cong \\Sym^2 V_4$. Adding the middle row of \\eqref{snake1-12} to $\\mathcal{O} \\rightarrow \\mathcal{O}$, we get\n\\[\n0 \\rightarrow \\mathcal{O}(-2)\\oplus \\mathcal{O} \\rightarrow \\mathcal{O}^{\\oplus 11} \\rightarrow \\Lambda \\rightarrow 0,\n\\]\nwhence the conclusion of the lemma.\n\\end{proof}\n\\end{lemma}\n\nThe previous lemma yields that a model for \\hyperlink{Fano1--12}{1--12} is $\\mathscr{Z}(\\Lambda(0,1) \\oplus \\mathcal{O}(0,2)) \\subset \\mathbb{P}^3 \\times \\mathbb{P}^{10}$.\n\n\n\\hypertarget{Fano2--2}{\\subsection*{Fano 2--2}}\n\\subsubsection*{Mori-Mukai} Double cover of $\\mathbb{P}^1 \\times \\mathbb{P}^2$ with branch locus a $(2,4)$ divisor.\n\\subsubsection*{Our description} $\\mathscr{Z} (\\mathcal{O}(0,0,2) \\oplus K(0,0,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^{12}$, where $K \\in \\Ext^2(\\mathcal{O}(1,0)^{\\oplus 6},\\mathcal{Q}_{\\mathbb{P}^2}(-1,-1))$ fits into sequences \\eqref{K2-2}.\n\\subsubsection*{Identification}\nBy Lemma \\ref{lem:doublecovers}, our Fano variety is the zero locus of $\\mathcal{O}(2)$ over $\\mathbb{P}(\\mathcal{O}(-1,-2) \\oplus \\mathcal{O})$, the latter being a projective bundle on $\\mathbb{P}^1 \\times \\mathbb{P}^2$. We need to express such projective bundle as the zero locus of a suitable vector bundle. \n\n\\begin{lemma}\n\\label{projBundle2-2}\nThe projective bundle $\\mathbb{P}(\\mathcal{O}(-1,-2) \\oplus \\mathcal{O})$ can be obtained as the zero locus of $K(0,0,1)$ over $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^{12}$, being $K \\in \\Ext^2(\\mathcal{O}(1,0)^{\\oplus 6},\\mathcal{Q}_{\\mathbb{P}^2}(-1,-1))$ a uniquely defined extension on $\\mathbb{P}^1 \\times \\mathbb{P}^2$ fitting into sequences \\eqref{K2-2} below.\n\\begin{proof}\nOur goal is to write $\\mathcal{O}_{\\mathbb{P}^1 \\times \\mathbb{P}^2}(-1,-2) \\oplus \\mathcal{O}_{\\mathbb{P}^1 \\times \\mathbb{P}^2}$ as a subbundle of $\\mathcal{O}_{\\mathbb{P}^1 \\times \\mathbb{P}^2}^{\\oplus 13}$. By Remark \\ref{rem:principalParts}, we have two (dual) canonical short exact sequences on $\\mathbb{P}^2$\n\\begin{gather*}\n0 \\rightarrow \\mathcal{O}(-2) \\rightarrow \\mathcal{O}(-1)^{\\oplus 3} \\rightarrow \\mathcal{Q}(-1) \\rightarrow 0,\\\\\n0 \\rightarrow \\mathcal{O}(-1)^{\\oplus 3} \\rightarrow \\mathcal{O}^{\\oplus 6} \\rightarrow \\Sym^2 \\mathcal{Q} \\rightarrow 0.\n\\end{gather*}\nThese fit as the first row and middle column of the exact diagram on $\\mathbb{P}^2$ here below, which can be completed by the snake lemma as\n\\begin{equation}\n\\label{snake2-2}\n\\begin{gathered}\n\\xymatrix{\n& 0 \\ar[d] & 0 \\ar[d] & 0 \\ar[d]\\\\\n0 \\ar[r] & \\mathcal{O}(-2) \\ar[d]^-= \\ar[r] & \\mathcal{O}(-1)^{\\oplus 3} \\ar[d] \\ar[r] & \\mathcal{Q}(-1) \\ar[d] \\ar[r] & 0 \\\\\n0 \\ar[r] & \\mathcal{O}(-2) \\ar[d] \\ar[r] & \\mathcal{O}^{\\oplus 6} \\ar[d] \\ar[r] & \\Lambda \\ar[d] \\ar[r] & 0 \\\\\n0 \\ar[r] & 0 \\ar[r]\\ar[d] & \\Sym^2 \\mathcal{Q} \\ar[r]^-= \\ar[d] & \\Sym^2 \\mathcal{Q} \\ar[r]\\ar[d] & 0\\\\\n& 0 & 0 & 0 \n}\n\\end{gathered}\n\\end{equation}\nfor a uniquely determined homogeneous vector bundle $\\Lambda$ of rank $5$. The last column describes $\\Lambda$ as a non-split extension\n\\begin{equation}\n\\label{LambdaProvv}\n0 \\rightarrow \\mathcal{Q}_{\\mathbb{P}^2}(-1) \\rightarrow \\Lambda \\rightarrow \\Sym^2 \\mathcal{Q}_{\\mathbb{P}^2} \\rightarrow 0.\n\\end{equation}\n\nWe can pull back the middle row of \\eqref{snake2-2} on $\\mathbb{P}^1 \\times \\mathbb{P}^2$ and twist it by $\\mathcal{O}(-1,0)$. This and the standard (pulled back) Euler sequence on $\\mathbb{P}^1$ can be inserted as the first row and second column in the exact diagram below, which can be again completed by the snake lemma as\n\\begin{equation}\n\\label{KSnake}\n\\begin{gathered}\n\\xymatrix{\n& 0 \\ar[d] & 0 \\ar[d] & 0 \\ar[d]\\\\\n0 \\ar[r] & \\mathcal{O}(-1,-2) \\ar[d]^-= \\ar[r] & \\mathcal{O}(-1,0)^{\\oplus 6} \\ar[d] \\ar[r] & \\Lambda(-1,0) \\ar[d] \\ar[r] & 0 \\\\\n0 \\ar[r] & \\mathcal{O}(-1,-2) \\ar[d] \\ar[r] & \\mathcal{O}^{\\oplus 12} \\ar[d] \\ar[r] & K \\ar[d] \\ar[r] & 0 \\\\\n0 \\ar[r] & 0 \\ar[r]\\ar[d] & \\mathcal{O}(1,0)^{\\oplus 6} \\ar[r]^-= \\ar[d] & \\mathcal{O}(1,0)^{\\oplus 6} \\ar[r]\\ar[d] & 0 \\\\\n& 0 & 0 & 0 \\\\\n}\n\\end{gathered}\n\\end{equation}\nfor a uniquely determined homogeneous vector bundle $K$ of rank $11$. We can further describe $K$ as an element of $\\Ext^2(\\mathcal{O}(1,0)^{\\oplus 6},\\mathcal{Q}_{\\mathbb{P}^2}(-1,-1))$ obtained by combining the short exact sequences\n\\begin{equation}\n\\begin{gathered}\n\\label{K2-2}\n0 \\rightarrow \\mathcal{Q}_{\\mathbb{P}^2}(-1,-1) \\rightarrow \\Lambda(-1,0) \\rightarrow \\Sym^2 \\mathcal{Q}_{\\mathbb{P}^2}(-1,0) \\rightarrow 0,\n\\\\\n0 \\rightarrow \\Lambda(-1,0) \\rightarrow K \\rightarrow \\mathcal{O}(1,0)^{\\oplus 6} \\rightarrow 0.\n\\end{gathered}\n\\end{equation}\n\nThe bundle $K$ is homogeneous, not completely reducible and globally generated, and its space of global sections coincides with $H^0(\\mathbb{P}^1 \\times \\mathbb{P}^2, \\mathcal{O}(1,0)^{\\oplus 6})$. Adding the middle row of \\eqref{KSnake} to $\\mathcal{O} \\rightarrow \\mathcal{O}$, we get\n\\[\n0 \\rightarrow \\mathcal{O}(-1,-2)\\oplus \\mathcal{O} \\rightarrow \\mathcal{O}^{\\oplus 13} \\rightarrow K \\rightarrow 0,\n\\]\nwhence the conclusion of the lemma.\n\\end{proof}\n\\end{lemma}\n\nBy construction, the bundle $\\mathcal{O}(2)$ on $\\mathbb{P}(\\mathcal{O}(-1,-2) \\oplus \\mathcal{O})$ is identified with $\\mathcal{O}(0,0,2)$ over the zero locus of $K$ on $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^{12}$, whence the conclusion. \n\n\n\\hypertarget{Fano2--3}{\\subsection*{Fano 2--3}}\n\\subsubsection*{Mori-Mukai} Blow up of \\hyperlink{Fano1--12}{1--12} in an elliptic curve which is the intersection of two divisors from $|-\\frac{1}{2}K|$.\n\\subsubsection*{Our description} $\\mathscr{Z} (\\mathcal{O}(4,0) \\oplus \\mathcal{O}(1,1)) \\subset \\mathbb{P}(1,1,1,1,2) \\times \\mathbb{P}^1$.\n\n\\subsubsection*{Identification} The first bundle on $\\mathbb{P}(1,1,1,1,2)$ gives \\hyperlink{Fano1--12}{1--12}. We can conclude by Lemma \\ref{lem:blowup}.\n\nIt is possible to provide a rather simple description involving only projective spaces. To do this, recall that a model for \\hyperlink{Fano1--12}{1--12} is $\\mathscr{Z}(\\Lambda(0,1) \\oplus \\mathcal{O}(0,2)) \\subset \\mathbb{P}^3 \\times \\mathbb{P}^{10}$. The adjunction formula shows that the canonical divisor is the restriction of $\\mathcal{O}(-2,0)$; by Lemma \\ref{lem:blowup}, a model for \\hyperlink{Fano2--3}{2--3} is therefore given by\n\\[\n\\mathscr{Z}(\\Lambda(0,1,0) \\oplus \\mathcal{O}(0,2,0) \\oplus \\mathcal{O}(1,0,1)) \\subset \\mathbb{P}^3 \\times \\mathbb{P}^{10} \\times \\mathbb{P}^1.\n\\]\n\n\\hypertarget{Fano2--5}{\\subsection*{Fano 2--5}}\n\\subsubsection*{Mori-Mukai} Blow up of 1--13 in a plane cubic.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(0,3) \\oplus \\mathcal{O}(1,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^4$.\n\\subsubsection*{Identification} The first bundle on $\\mathbb{P}^4$ gives 1--13. We conclude by Lemma \\ref{lem:blowup}.\n\n\\hypertarget{Fano2--8}{\\subsection*{Fano 2--8}}\n\\subsubsection*{Mori-Mukai} Double cover of \\hyperlink{Fano2--35}{2--35} with branch locus an anticanonical divisor such that the intersection with the exceptional divisor is smooth.\n\n\\subsubsection*{Our description} $\\mathscr{Z}(\\Lambda(0,0,1) \\oplus \\mathcal{O}(0,0,2)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^3 \\times \\mathbb{P}^{12}$, being $\\Lambda \\in \\Ext^1(\\mathcal{Q}_{\\mathbb{P}^3}^{\\oplus 3},\\mathcal{Q}_{\\mathbb{P}^2}(0,-1))$ a uniquely defined extension on $\\mathbb{P}^2 \\times \\mathbb{P}^3$ fitting into sequence \\eqref{Lambda2-8} below.\n\\subsubsection*{Identification}\nAs shown below, $Y:={}$ \\hyperlink{Fano2--35}{2--35} can be obtained as $\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^2}(0,1)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^3$. By Lemma \\ref{lem:doublecovers}, our Fano variety is the zero locus of $\\mathcal{O}(2)$ on $\\mathbb{P}_Y(\\mathcal{O}(-1,-1) \\oplus \\mathcal{O})$.\n\nAs it turns out, the projective bundle $\\mathbb{P}_{\\mathbb{P}^2 \\times \\mathbb{P}^3}(\\mathcal{O}(-1,-1) \\oplus \\mathcal{O})$ can be obtained as the zero locus of $\\Lambda(0,0,1)$ over $\\mathbb{P}^2 \\times \\mathbb{P}^3 \\times \\mathbb{P}^{12}$, being $\\Lambda \\in \\Ext^1(\\mathcal{Q}_{\\mathbb{P}^3}^{\\oplus 3},\\mathcal{Q}_{\\mathbb{P}^2}(0,-1))$ a uniquely defined extension on $\\mathbb{P}^2 \\times \\mathbb{P}^3$ fitting into sequence \\eqref{Lambda2-8} below. To see it, we can argue as in Lemma \\ref{projBundle1-12} or Lemma \\ref{projBundle2-2}: we combine the (pull back of the) two (possibly twisted) Euler sequences\n\\begin{gather*}\n0 \\rightarrow \\mathcal{O}(-1,-1) \\rightarrow \\mathcal{O}(0,-1)^{\\oplus 3} \\rightarrow \\mathcal{Q}_{\\mathbb{P}^2}(0,-1) \\rightarrow 0,\\\\\n0 \\rightarrow \\mathcal{O}(0,-1)^{\\oplus 3} \\rightarrow \\mathcal{O}^{\\oplus 12} \\rightarrow \\mathcal{Q}_{\\mathbb{P}^3}^{\\oplus 3} \\rightarrow 0.\n\\end{gather*}\nWe get\n\\begin{gather}\n\\label{inclusion2-8}\n0 \\rightarrow \\mathcal{O}(-1,-1) \\rightarrow \\mathcal{O}^{\\oplus 12} \\rightarrow \\Lambda \\rightarrow 0,\n\\\\\n\\label{Lambda2-8}\n0 \\rightarrow \\mathcal{Q}_{\\mathbb{P}^2}(0,-1) \\rightarrow \\Lambda \\rightarrow \\mathcal{Q}_{\\mathbb{P}^3}^{\\oplus 3} \\rightarrow 0,\n\\end{gather}\nwhere the rank $11$ bundle $\\Lambda$ is homogeneous, not completely reducible and globally generated, and its space of global sections coincides with $H^0(\\mathbb{P}^3, \n\\mathcal{Q}^{\\oplus 3}) \\cong (V_4)^{\\oplus 3}$. Adding $\\mathcal{O} \\rightarrow \\mathcal{O}$ to \\eqref{inclusion2-8} we get the desired description for $\\mathbb{P}_{\\mathbb{P}^2 \\times \\mathbb{P}^3}(\\mathcal{O}(-1,-1) \\oplus \\mathcal{O})$ and the conclusion.\n\n\n\\hypertarget{Fano2--10}{\\subsection*{Fano 2--10}}\n\\subsubsection*{Mori-Mukai} Blow up of 1--14 in an elliptic curve which is an intersection of 2 hyperplanes.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(2,0) \\oplus \\mathcal{O}(1,1))\\subset \\Gr(2,4) \\times \\mathbb{P}^1$.\n\\subsubsection*{Identification} See Lemma \\ref{lem:blowup}.\n\n\\hypertarget{Fano2--11}{\\subsection*{Fano 2--11}}\n\\subsubsection*{Mori-Mukai} Blow up of 1--13 in a line.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^2}(0,1) \\oplus \\mathcal{O}(1,2)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^4$.\n\\subsubsection*{Identification} By Lemma \\ref{lem:blow}, the zero locus of the first summand gives $\\Bl_{\\mathbb{P}^1}(\\mathbb{P}^4)$. Let $\\mathbb{P}^4=\\mathbb{P}(V_5)$ with dual coordinates $x_0, \\dotsc, x_4 \\in V_5^\\vee$. Assume that $\\mathbb{P}^1=\\mathbb{P}(V_2)$ is given by the vanishing of $x_2, \\dotsc, x_4$. A general section in $H^0(\\mathbb{P}^2 \\times \\mathbb{P}^4,\\mathcal{O}(1,2))$ is identified with a cubic in $\\Sym^3(V_5^\\vee)\/\\Sym^3(V_2^\\vee)$, i.e., a cubic without terms in $x_0^3, x_0^2x_1, x_0x_1^2,x_1^3$. Such cubic contains $\\mathbb{P}(V_2)$, hence the claim.\nNotice that, using the equivalent Corollary \\ref{cor:blowupflag}, we can describe \\hyperlink{Fano2--11}{2--11} as well as the zero locus $\\mathscr{Z}(\\mathcal{Q}_2^{\\oplus 2} \\oplus \\mathcal{O}(2,1)) \\subset \\Fl(1,3,5)$.\n\n\n\n\n\\hypertarget{Fano2--15}{\\subsection*{Fano 2--15}}\n\\subsubsection*{Mori-Mukai}Blow up of $\\mathbb{P}^3$ in the intersection of a quadric and a cubic where the quadric is smooth.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^3}(0,1) \\oplus \\mathcal{O}(2,1)) \\subset \\mathbb{P}^3 \\times \\mathbb{P}^4$.\n\\subsubsection*{Identification}\n\nBy Lemma \\ref{lem:blowDegeneracyLocus}, our Fano is the zero locus of $\\mathcal{O}(1) \\otimes \\pi^*\\mathcal{O}(2)$ over $\\pi:\\mathbb{P}(\\mathcal{O}(-1)\\oplus \\mathcal{O}) \\rightarrow \\mathbb{P}^3$.\n\nAdding $\\mathcal{O} \\rightarrow \\mathcal{O}$ to the standard Euler sequence on $\\mathbb{P}^3$ we get\n\\[\n0 \\rightarrow\n\\mathcal{O}(-1) \\oplus \\mathcal{O} \\rightarrow\n\\mathcal{O}^{\\oplus 5} \\rightarrow\n\\mathcal{Q} \\rightarrow\n0,\n\\]\nwhence the result.\n\nAnother simple description of our Fano is\n\\begin{equation}\n\\label{anotherDescription}\n\\mathscr{Z}(\\mathcal{Q}_2 \\oplus \\mathcal{O}(1,2)) \\subset \\Fl(1,2,5);\n\\end{equation}\nthe two descriptions are equivalent thanks to the correspondence between subvarieties of flags and of products of Grassmannians given by Lemma \\ref{lem:identificationsOnFlags}, Remark \\ref{rmk:wisniewski}, and Lemma \\ref{lem:blow}. From these one can immediately identify $(\\mathscr{Z}(\\mathcal{Q}_2) \\subset \\Fl(1,2,V_5)) \\cong \\mathbb{P}_{\\mathbb{P}(V_5\/v_0)}(\\mathcal{O}(-1) \\oplus (v_0 \\otimes\\mathcal{O})) \\cong \\Bl_{\\mathbb{P}(v_0)} \\mathbb{P}(V_5)$ as $\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^3}\\boxtimes \\mathcal{O}_{\\mathbb{P}^4}(1)) \\subset \\mathbb{P}(V_5\/v_0) \\times \\mathbb{P}((V\/v_0) \\oplus v_0)$. On the latter we have that $\\mathcal{O}(1,0) \\cong p^*\\mathcal{O}_{\\mathbb{P}^3}(1)$ and $\\mathcal{O}(0,1) \\cong \\pi^*\\mathcal{O}_{\\mathbb{P}^4}(1)$, where $p$ is the projective bundle map and $\\pi$ the blow up map.\n\nWe want to provide a direct way to describe our Fano as \\eqref{anotherDescription}, in order to show how the Cayley trick can be effectively used. First note that by Corollary \\ref{cor:onF12n} $X=\\mathscr{Z}(\\mathcal{Q}_2) \\subset F:=\\Fl(1,2,5)$ is identified with $\\mathbb{P}(E)=\\mathbb{P}_{\\mathbb{P}^3}(\\mathcal{O}(-1) \\oplus \\mathcal{O}) \\cong \\mathbb{P}_{\\mathbb{P}^3}(\\mathcal{O}(-3) \\oplus \\mathcal{O}(-2)) $. To use Corollary \\ref{cor:cayleycrit} we want to show that \n\\[\nH^0(X, \\mathcal{O}_X(1,2)) \\cong H^0(\\mathbb{P}(E), \\mathcal{O}_{\\mathbb{P}(E)}(1)) \\cong H^0(\\mathbb{P}^3, \\mathcal{O}_{\\mathbb{P}^3}(2) \\oplus \\mathcal{O}_{\\mathbb{P}^3}(3)).\n\\]\nIn order to compute $H^0(X, \\mathcal{O}_X(1,2))$ we use the Koszul complex for $X \\subset F$ twisted by $\\mathcal{O}_{F}(1,2)$. The only non-zero cohomology groups are\n\\[\n\\begin{array}{ll}\nH^0(F, \\bigwedge^3 \\mathcal{Q}^{\\vee}_2 \\otimes \\mathcal{O}_F(1,2)) \\cong \\Sigma_{2,2,2,1}V_5 \\cong \\mathbb{C}^{40},&\nH^0(F, \\bigwedge^2 \\mathcal{Q}^{\\vee}_2 \\otimes \\mathcal{O}_F(1,2)) \\cong \\Sigma_{3,2,2,1}V_5 \\cong \\mathbb{C}^{175}\\\\\n\\rule{0pt}{12pt}\nH^0(F, \\mathcal{Q}^{\\vee}_2 \\otimes \\mathcal{O}_F(1,2)) \\cong \\Sigma_{3,3,2,1}V_5 \\cong \\mathbb{C}^{280},&\nH^0(F, \\mathcal{O}_F(1,2)) \\cong \\Sigma_{3,3,3,1}V_5 \\cong \\mathbb{C}^{175}.\n\\end{array}\n\\]\nAs in Lemma \\ref{lem:identificationsOnFlags}, $\\mathcal{U}_2|_X= \\overline{\\mathcal{U}_1} \\oplus \\mathcal{O}$: this is therefore equivalent to split $V_5 = V_4 \\oplus \\mathbb{C} v_0$, and apply the above Schur functors to a such decomposed $V_5$ to get $\\SL(4)\\times \\mathbb{C}^*$ representations, with the $\\mathbb{C}^*$ component being the trivial representation. As it turns out,\n\\[\n\\begin{array}{rcl}\n\\Sigma_{2,2,2,1}(V_4 \\oplus \\mathbb{C}) & = & \\Sigma_{2,2,1}V_4\\oplus\\Sigma_{2,2,2}V_4\\oplus\\Sigma_{2,2,1,1}V_4\\oplus\\Sigma_{2,2,2,1}V_4, \\\\\n\\rule{0pt}{12pt} \\Sigma_{3,2,2,1}(V_4 \\oplus \\mathbb{C}) & = & \\Sigma_{3,2,2,1}V_4\\oplus\\Sigma_{3,2,2}V_4\\oplus\\Sigma_{3,2,1,1}V_4\\oplus\\Sigma_{3,2,1}V_4\\oplus\\Sigma_{2,2,2,1}V_4\\oplus\\Sigma_{2,2,2}V_4\\oplus\\\\ & & {}\\oplus \\Sigma_{2,2,1,1}V_4\\oplus\\Sigma_{2,2,1}V_4,\\\\\n\\rule{0pt}{12pt} \\Sigma_{3,3,2,1}(V_4 \\oplus \\mathbb{C})& = &\\Sigma_{3,3,2,1}V_4\\oplus\\Sigma_{3,3,2}V_4\\oplus\\Sigma_{3,3,1,1}V_4\\oplus\\Sigma_{3,3,1}V_4\\oplus\\Sigma_{3,2,2,1}V_4\\oplus \\Sigma_{3\n ,2,2}V_4\\oplus\\\\ & &{}\\oplus\\Sigma_{3,2,1,1}V_4\\oplus\\Sigma_{3,2,1}V_4,\\\\\n\\rule{0pt}{12pt} \\Sigma_{3,3,3,1}(V_4 \\oplus \\mathbb{C})& = &\\Sigma_{3,3,3,1}V_4\\oplus\\Sigma_{3,3,3}V_4\\oplus\\Sigma_{3,3,2,1}V_4\\oplus\\Sigma_{3,3,2}V_4\\oplus\\Sigma_{3,3,1,1}V_4\\oplus\\Sigma_{3\n ,3,1}V_4.\n\\end{array}\n\\]\n\n\nTherefore, splitting the Koszul complex in short exact sequences, we get the natural isomorphism (of vector spaces)\n\\[H^0(X, \\mathcal{O}_X(1,2)) \\cong \\Sigma_{2,2,2}V_4 \\oplus \\Sigma_{3,3,3}V_4 \\cong \\Sym^2 V_4^{\\vee} \\oplus \\Sym^3 V_4^{\\vee} \\cong H^0(\\mathbb{P}^3, \\mathcal{O}_{\\mathbb{P}^3}(2) \\oplus \\mathcal{O}_{\\mathbb{P}^3}(3)),\\]\nas claimed. It suffices to use Corollary \\ref{cor:cayleycrit} to show that $X$ coincides with the Mori--Mukai description as the blow up of $\\mathbb{P}^3$ in the complete intersection of a quadric and a cubic surfaces.\n\n\n\\hypertarget{Fano2--16}{\\subsection*{Fano 2--16}}\n\\subsubsection*{Mori-Mukai} Blow up of 1--14 in a conic.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,0) \\oplus \\mathcal{O}(0,2)) \\subset \\Fl(1,2,4)$.\n\\subsubsection*{Identification} Let $Y=\\mathscr{Z}(\\mathcal{O}_F(0,2)) \\subset \\Fl(1,2,4)$ and $\\widetilde{Y}= \\mathscr{Z}(\\mathcal{O}_G(2)) \\subset \\Gr(2,4)$. One directly checks that \\[H^0(Y, \\mathcal{O}_Y(1,0)) \\cong H^0(\\widetilde{Y}, \\overline{\\mathcal{U}}^{\\vee}|_{\\widetilde{Y}}).\\]\nIn fact, both spaces can be naturally identified with $V_4^{\\vee}$, as in \\hyperlink{Fano2--15}{2--15}. Then it suffices to apply Corollary \\ref{cor:cayleycrit} to get that $X=\\mathscr{Z}(\\mathcal{O}_Y(1,0)) \\subset Y \\cong \\Bl_{\\mathscr{Z}( \\overline{\\mathcal{U}}^{\\vee}|_{\\widetilde{Y}})}\\widetilde{Y}$, where we used that by duality on $\\Gr(2,4)$, $\\mathcal{U}(1) \\cong \\mathcal{U}^{\\vee} \\cong (\\pi_2)_{*} \\mathcal{O}_F(1,0)$. We conclude the proof by noting that $\\widetilde{Y}$ is a complete intersection of two quadrics in $\\mathbb{P}^5$, and $(\\mathscr{Z}( \\overline{\\mathcal{U}}^{\\vee}|_{\\widetilde{Y}}) \\subset \\widetilde{Y}) = \\mathscr{Z}(\\mathcal{U}^{\\vee} \\oplus \\mathcal{O}_G(2)) \\subset \\Gr(2,4)$ which is a plane conic.\n\nWe want to give an alternative description of this Fano in the product of two Grassmannians. For this, let us start by the Mori--Mukai description. Lemma \\ref{lem:blowInGrass} enables us to describe $\\Bl_{\\mathbb{P}^2} \\Gr(2,4)$ in the product $(\\mathbb{P}^2)^{\\vee} \\times \\Gr(2,4)$. We then need to cut with an extra quadric intersecting the blown up $\\mathbb{P}^2$. As we are going to see in full details for \\hyperlink{Fano2--26}{2--26}, for this it suffices to take a section of $\\mathcal{O}(0,2)$. Summarising, we can describe our \\hyperlink{Fano2--16}{2--16} as \n\\[ \\mathscr{Z}(\\mathcal{U}^{\\vee}_{\\Gr(2,4)}(1,0) \\oplus \\mathcal{O}(0,2)) \\subset \\mathbb{P}^2 \\times \\Gr(2,4).\n\\]\n\\hypertarget{Fano2--17}{\\subsection*{Fano 2--17}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{Q}_3$ in an elliptic curve of degree 5.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(0,1) \\oplus \\mathcal{O}(1,1)) \\subset \\Fl(1,2,4)$.\n\\subsubsection*{Identification}\nA model for this 3-fold in $\\Gr(2,4) \\times \\mathbb{P}^3$ can be found in \\cite{corti}. As an exception to our self-imposed rule, we want to give here an alternative description in a flag variety, since we find it particularly nice. Let us show that our Fano can be written as\n\\[\n\\mathscr{Z}(\\mathcal{O}(1,1) \\oplus \\mathcal{O}(0,1)) \\subset \\Fl(1,2,4).\n\\]\n\n As before, we check that \\[H^0(Y, \\mathcal{O}_Y(1,1)) \\cong H^0(\\widetilde{Y}, \\overline{\\mathcal{U}}^{\\vee}(1)|_{\\widetilde{Y}}),\\]\nwhere we are using the same notation as above. These spaces are both $16$-dimensional and isomorphic as vector spaces to $\\Sigma_{2,1}V_4^{\\vee}\/V_4^{\\vee}$, where we can interpret $\\Sigma_{2,1}V_4$ as the kernel of the natural contraction map $\\lrcorner: V_4 \\otimes \\bigwedge^2 V_4^{\\vee} \\rightarrow V_4^{\\vee}$. These spaces of sections are not $\\SL(V_4)$-representations: in fact $\\widetilde{Y}$ (and similarly for the section on the flag) is not homogeneous for the whole group, but rather for $\\SO(V_3)$, and one could write a more elegant expression for the spaces of section as in \\hyperlink{Fano2--15}{2--15}. To conclude we apply Corollary \\ref{cor:cayleycrit}: we have that $X=\\mathscr{Z}(\\mathcal{O}_Y(1,1)) \\cong \\Bl_{\\widetilde{Z}} \\widetilde{Y}$ where $\\widetilde{Y}$ is a quadric 3-fold, and the centre of the blow up is $\\widetilde{Z}= \\mathscr{Z}(\\mathcal{U}^{\\vee}(1) \\oplus \\mathcal{O}(1)) \\subset \\Gr(2,4)$, which can be easily checked to be an elliptic curve of degree 5.\n\n\\hypertarget{Fano2--18}{\\subsection*{Fano 2--18}}\n\\subsubsection*{Mori-Mukai} Double cover of 2-34 with branch locus a divisor of degree $(2,2)$.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\Lambda(0,0,1) \\oplus \\mathcal{O}(0,0,2)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^6$, being $\\Lambda \\in \\Ext^1(\\mathcal{Q}_{\\mathbb{P}^2}^{\\oplus 2},\\mathcal{O}(1,-1))$ a uniquely defined extension on $\\mathbb{P}^1 \\times \\mathbb{P}^2$ fitting into sequence \\eqref{Lambda2-18} below.\n\n\\subsubsection*{Identification}\n\nBy Lemma \\ref{lem:doublecovers}, our Fano variety is the zero locus of $\\mathcal{O}(2)$ on $\\mathbb{P}_{\\mathbb{P}^1 \\times \\mathbb{P}^2}(\\mathcal{O}(-1,-1) \\oplus \\mathcal{O})$. As it turns out, the latter projective bundle can be obtained as the zero locus of $\\Lambda(0,0,1)$ over $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^6$, being $\\Lambda \\in \\Ext^1(\\mathcal{Q}_{\\mathbb{P}^2}^{\\oplus 2},\\mathcal{O}(1,-1))$ a uniquely defined extension on $\\mathbb{P}^1 \\times \\mathbb{P}^2$ fitting into sequence \\eqref{Lambda2-18} below. To see it, we can argue as in Lemma \\ref{projBundle1-12} or Lemma \\ref{projBundle2-2}: we combine the (pull back of the) two (possibly twisted) Euler sequences\n\\begin{gather*}\n0 \\rightarrow \\mathcal{O}(-1,-1) \\rightarrow \\mathcal{O}(0,-1)^{\\oplus 2} \\rightarrow \\mathcal{O}(1,-1) \\rightarrow 0,\\\\\n0 \\rightarrow \\mathcal{O}(0,-1)^{\\oplus 2} \\rightarrow \\mathcal{O}^{\\oplus 6} \\rightarrow \\mathcal{Q}_{\\mathbb{P}^2}^{\\oplus 2} \\rightarrow 0.\n\\end{gather*}\nWe get\n\\begin{gather}\n\\label{inclusion2-18}\n0 \\rightarrow \\mathcal{O}(-1,-1) \\rightarrow \\mathcal{O}^{\\oplus 6} \\rightarrow \\Lambda \\rightarrow 0,\n\\\\\n\\label{Lambda2-18}\n0 \\rightarrow \\mathcal{O}(1,-1) \\rightarrow \\Lambda \\rightarrow \\mathcal{Q}_{\\mathbb{P}^2}^{\\oplus 2} \\rightarrow 0,\n\\end{gather}\nwhere the rank $5$ bundle $\\Lambda$ is homogeneous, not completely reducible and globally generated, and its space of global sections coincides with $H^0(\\mathbb{P}^2, \n\\mathcal{Q}^{\\oplus 2}) \\cong (V_3)^{\\oplus 2}$. Adding $\\mathcal{O} \\rightarrow \\mathcal{O}$ to \\eqref{inclusion2-18} we get the desired description for $\\mathbb{P}_{\\mathbb{P}^1 \\times \\mathbb{P}^2}(\\mathcal{O}(-1,-1) \\oplus \\mathcal{O})$ and the conclusion.\n\n\\hypertarget{Fano2--19}{\\subsection*{Fano 2--19}}\n\\subsubsection*{Mori-Mukai} Blow up of 1--14 in a line.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^3}(0,1) \\oplus \\mathcal{O}(1,1)^{\\oplus 2} ) \\subset \\mathbb{P}^3 \\times \\mathbb{P}^5$. \n\n\\subsubsection*{Identification} It suffices to apply Lemma \\ref{lem:blow} and argue as done for \\hyperlink{Fano2--11}{2--11}. The zero locus of the first factor identifies $\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^3}(0,1))$ with the blow up $\\Bl_{\\mathbb{P}^1}(\\mathbb{P}^5)$. Let $\\mathbb{P}^5=\\mathbb{P}(V_6)$ with dual coordinates $x_0, \\dotsc, x_5 \\in V_6^\\vee$. Assume that $\\mathbb{P}^1=\\mathbb{P}(V_2)$ is given by the vanishing of $x_2, \\dotsc, x_5$. A general section in $H^0(\\mathbb{P}^3 \\times \\mathbb{P}^5,\\mathcal{O}(1,1)^2)$ is identified with two quadrics in $\\Sym^2(V_6^\\vee)\/\\Sym^2(V_2^\\vee)$, i.e., quadrics without terms in $x_0^2, x_1^2, x_0x_1$. Such quadrics have generically maximal rank, so their intersection is smooth and contains $\\mathbb{P}(V_2)$, hence the claim.\nNotice that, using the equivalent Corollary \\ref{cor:blowupflag}, we can describe \\hyperlink{Fano2--19}{2--19} as the zero locus of \n$\\mathscr{Z}(\\mathcal{Q}_2^{\\oplus 2} \\oplus \\mathcal{O}(1,1)^{\\oplus 2}) \\subset \\Fl(1,3,6)$ as well.\n\n\n\n\\hypertarget{Fano2--22}{\\subsection*{Fano 2--22}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{V}_5$ in a conic.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_{\\Gr(2,5)}(1,0) \\oplus \\mathcal{O}(0,1)^{\\oplus 3}) \\subset \\mathbb{P}^3 \\times \\Gr(2,5).$\n\\subsubsection*{Identification} In \\cite{corti} this variety is described as $\\mathscr{Z}(\\mathcal{O}(1,0) \\oplus \\mathcal{O}(0,1)^{\\oplus 3}) \\subset \\Fl(1,2,5)$. This description is equivalent to the one given here simply applying Lemma \\ref{lem:blowInGrass} with $k=3$ (where we identify $\\Gr(3,4)$ and $\\Gr(3,5)$ with $\\mathbb{P}^3$ and $\\Gr(2,5)$). The three residual sections of $\\mathcal{O}(0,1)$ cut both $\\Gr(2,5)$ (in $\\mathbb{V}_5$) and $\\Gr(2,4)$ (in a conic).\n\n\n\\hypertarget{Fano2--23}{\\subsection*{Fano 2--23}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{Q}_3$ in an intersection of a hyperplane and a quadric.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_2 \\oplus \\mathcal{O}(1,1) \\oplus \\mathcal{O}(0,2)) \\subset \\Fl(1,2,6)$.\n\n\\subsubsection*{Identification} We apply Corollary \\ref{cor:cayleycrit}. In the notation of the corollary, we denote by $Y \\subset \\Fl(1,2,6)$ the zero locus of $\\mathcal{Q}_2 \\oplus \\mathcal{O}(0,2)$. We identify $\\widetilde{Y}$ with a three dimensional quadric $Q \\subset \\mathbb{P}^4$. What we need to check is \\[H^0(Y, \\mathcal{O}_Y(1,1)) \\cong H^0(Q, \\mathcal{O}_Q(1)) \\oplus H^0(Q, \\mathcal{O}_Q(2)). \\]\n\nTo verify this, one can argue as for \\hyperlink{Fano2--15}{2--15}: one can compute the $\\SL(V_6)$-representations arising from the Koszul complex resolving $\\mathcal{O}_Y(1,1)$. These representations, when seen as $\\SL(V_5)\\times \\mathbb{C}^*$-representations under the splitting $V_6 = V_5 \\oplus \\mathbb{C} v_0$, sum up to $\\Sigma_{1,1,1,1}V_5 \\oplus \\Sigma_{2,2,2,2}V_5\/ \\mathbb{C}$, which is clearly isomorphic to the right hand side.\n\nTherefore $X \\cong \\Bl_{\\widetilde{Z}} Q$, where $\\widetilde{Z}$ is given by the intersection of a quadratic and linear forms in $Q$.\n\nWe provide the following alternative description for this Fano:\n\\[ \\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^4}(0,1) \\oplus \\mathcal{O}(2,0) \\oplus \\mathcal{O}(1,1)) \\subset \\mathbb{P}^4 \\times \\mathbb{P}^5,\n\\]\nwhich can be shown to be equivalent to the previous one following the same lines of \\hyperlink{Fano2--15}{2--15}.\n\n\n\\hypertarget{Fano2--26}{\\subsection*{Fano 2--26}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{V}_5$ in a curve of genus 0.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_{2,4} \\boxtimes \\mathcal{U}^{\\vee}_{2,5} \\oplus \\mathcal{O}(1,0) \\oplus \\mathcal{O}(0,1)^{\\oplus 2}) \\subset \\Gr(2,4) \\times \\Gr(2,5).$\n\n\\subsubsection*{Identification} \nBy Lemma \\ref{lem:blowInGrass} we can identify $\\mathscr{Z}(\\mathcal{Q}_{2,4} \\boxtimes \\mathcal{U}^{\\vee}_{2,5}) \\subset \\Gr(2,4) \\times \\Gr(2,5)$ as the blow up $\\Bl_{\\mathbb{P}^3} \\Gr(2,5)=\\Gr(2,V_5)$, where $\\mathbb{P}^3$ is identified with $\\mathscr{Z}(\\mathcal{Q}) \\subset \\Gr(2,V_5)$, given by a vector $w \\in V_5^\\vee$.\n\nWithout loss of generality, we can assume $w=x_0$. We have a splitting $V_5^{\\vee}= x_0 \\oplus W_4$ that induces a splitting $\\bigwedge^2 V_5^{\\vee} = \\bigwedge^2 W_4 \\oplus x_0 \\wedge W_4$. For simplicity, let us fix a basis $x_0,\\dotsc,x_4$ of $V_5^{\\vee}$ and the corresponding dual basis $e_0,\\dotsc,e_4$ of $V_5$. The above $\\mathbb{P}^3$ is by definition described by the points in $\\Gr(2,5)$ of the form $e_0 \\wedge \\alpha$, where $\\alpha \\in \\langle e_1,\\dotsc,e_4 \\rangle$.\n\nBy construction, any $f \\in \\bigwedge^2 W_4=|\\mathcal{O}(1,0)|$ does not contain any summand of the form $x_0 \\wedge \\beta$, so that $f(e_0 \\wedge \\alpha)=0$. In other words, $f \\in \\Ann(\\mathbb{P}^3)$, hence its zero locus in $\\Bl_{\\mathbb{P}^3} \\Gr(2,5)$ contains the whole exceptional divisor and does not cut it. The two extra sections of $\\mathcal{O}(0,1)$ cut the exceptional divisor in a codimension two linear subspace. Therefore our zero locus can be seen as the blow up of $\\mathscr{Z}(\\mathcal{O}_{\\Gr(2,V_5)}^{\\oplus 3}(1))\\subset \\Gr(2,V_5)$ along $\\mathbb{P}^1 \\cong \\mathscr{Z}(\\mathcal{O}_{\\mathbb{P}^3}^{\\oplus 2}(1))\\subset \\mathbb{P}^3$.\n\n\n\n\n\nAnother description of this Fano is as $\\mathscr{Z}(\\mathcal{U}_1^{\\vee} \\oplus \\mathcal{O}(1,0) \\oplus \\mathcal{O}(0,1)^{\\oplus 2}) \\subset \\Fl(2,3,5)$. By Lemma \\ref{lem:identificationsOnFlags} this can be easily identified with the alternative description of this Fano given in \\cite{corti}.\n\n\n\\hypertarget{Fano2--28}{\\subsection*{Fano 2--28}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{P}^3$ in a plane cubic.\n\\subsubsection*{Our description} \n$\\mathscr{Z}(\\Lambda(0,1) \\oplus \\mathcal{O}(1,1)) \\subset \\mathbb{P}^3 \\times \\mathbb{P}^{10}$, being $\\Lambda \\in \\Ext^1(\\Sym^2 \\mathcal{Q},\\mathcal{Q}(-1))$ a uniquely defined extension on $\\mathbb{P}^3$ fitting into sequence \\eqref{Lambda1--12} above.\n\n\\subsubsection*{Identification}\nOur Fano variety is the blow up of $\\mathbb{P}^3$ along the intersection of two divisors of degree $1$ and $3$. By Lemma \\ref{lem:blowDegeneracyLocus}, it corresponds to the zero locus of $\\pi^*\\mathcal{O}_{\\mathbb{P}^3}(1) \\otimes \\mathcal{O}(1)$ over the projective bundle $\\pi:\\mathbb{P}(\\mathcal{O}(-2)\\oplus \\mathcal{O})\\rightarrow \\mathbb{P}^3$. We conclude by Lemma \\ref{projBundle1-12}.\n\n\n\\hypertarget{Fano2--29}{\\subsection*{Fano 2--29}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{Q}_3$ in a conic.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(0,2) \\oplus \\mathcal{O}(1,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^4$.\n\n\\subsubsection*{Identification} See Lemma \\ref{lem:blowup}.\n\n\n\\hypertarget{Fano2--30}{\\subsection*{Fano 2--30}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{P}^3$ in a conic.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_2 \\oplus \\mathcal{O}(1,1)) \\subset \\Fl(1,2,5)$.\n\n\\subsubsection*{Identification} We apply Corollary \\ref{cor:cayleycrit}. Following the notation of the corollary, we denote by $Y \\subset \\Fl(1,2,5)$ the zero locus of $\\mathcal{Q}_2$ and we identify $\\widetilde{Y}$ with a $\\mathbb{P}^3$. What we need to check is \\[H^0(Y, \\mathcal{O}_Y(1,1)) \\cong H^0(\\mathbb{P}^3, \\mathcal{O}_{\\mathbb{P}^3}(1)\\oplus \\mathcal{O}_{\\mathbb{P}^3}(2)). \\]\n\nTo verify this, one can argue as for \\hyperlink{Fano2--15}{2--15} or \\hyperlink{Fano2--23}{2--23}: the representations arising from the Koszul complex resolving $\\mathcal{O}_Y(1,1)$, when seen as $\\SL(V_4)\\times \\mathbb{C}^*$-representations, sum up to $\\Sigma_{1,1,1}V_4 \\oplus \\Sigma_{2,2,2}V_4$, which is clearly isomorphic to the right hand side.\n\nNotice that as an alternative description we can follow the same lines of \\hyperlink{Fano2--15}{2--15} and describe the Fano \\hyperlink{Fano2--30}{2--30} as \n\\[\n\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^3}(0,1) \\oplus \\mathcal{O}(1,1)) \\subset \\mathbb{P}^3 \\times \\mathbb{P}^4.\n\\]\n\\hypertarget{Fano2--31}{\\subsection*{Fano 2--31}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{Q}_3$ in a line.\n\n\\subsubsection*{Our description} $\\mathscr{Z}( \\mathcal{U}^{\\vee}_{\\Gr(2,4)}(1,0) \\oplus \\mathcal{O}(0,1)) \\subset \\mathbb{P}^2 \\times \\Gr(2,4) $.\n\\subsubsection*{Identification}\nWe may regard $\\mathbb{P}^2$ as $\\Gr(2,3)$, so that our Fano is given as $\\mathscr{Z}(\\mathcal{Q} \\boxtimes \\mathcal{U}^{\\vee} \\oplus \\mathcal{O}(0,1))$. Then we argue as for \\hyperlink{Fano2--26}{2--26}. By Lemma \\ref{lem:blowInGrass} we can identify $\\mathscr{Z}(\\mathcal{Q} \\boxtimes \\mathcal{U}^{\\vee}) \\subset \\Gr(2,3) \\times \\Gr(2,4)$ as $\\Bl_{\\mathbb{P}^2} \\Gr(2,4)$, where $\\mathbb{P}^2$ is identified with $\\mathscr{Z}(\\mathcal{Q}) \\subset \\Gr(2,4)$. As shown for \\hyperlink{Fano2--26}{2--26}, the remaining section of $\\mathcal{O}(0,1)$ cuts such $\\mathbb{P}^2$ in a codimension one linear subspace and the ambient $\\Gr(2,4)$ in a three-dimensional quadric, hence the conclusion.\n\n\n\\hypertarget{Fano2--33}{\\subsection*{Fano 2--33}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{P}^3$ in a line.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^3$.\n\\subsubsection*{Identification} See Lemma \\ref{lem:blowup}.\n\n\n\\hypertarget{Fano2--35}{\\subsection*{Fano 2--35}}\n\\subsubsection*{Mori-Mukai} $\\Bl_p \\mathbb{P}^3$ or $\\mathbb{P}_{\\mathbb{P}^2}(\\mathcal{O} \\oplus \\mathcal{O}(-1))$.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^2}(0,1)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^3$.\n\\subsubsection*{Identification}\nThis is a straightforward application of Lemma \\ref{lem:blow}. Notice that equivalently we could describe \\hyperlink{Fano2--35}{2--35} as $\\mathscr{Z}(\\mathcal{Q}_2) \\subset \\Fl(1,2,4)$.\n\n\\hypertarget{Fano2--36}{\\subsection*{Fano 2--36}}\n\\subsubsection*{Mori-Mukai} $\\mathbb{P}_{\\mathbb{P}^2}(\\mathcal{O} \\oplus \\mathcal{O}(-2))$.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\Lambda(0,1)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^6$, being $\\Lambda \\in \\Ext^1(\\Sym^2 \\mathcal{Q},\\mathcal{Q}(-1))$ a uniquely defined extension on $\\mathbb{P}^2$ fitting into sequence \\eqref{Lambda2-36}.\n\n\\subsubsection*{Identification}\nWe argue as in Lemma \\ref{projBundle1-12}, with the appropriate changes. By Remark \\ref{rem:principalParts}, we have two (dual) canonical short exact sequences on $\\mathbb{P}^2$\n\\begin{gather*}\n0 \\rightarrow \\mathcal{O}(-2) \\rightarrow 3\\mathcal{O}(-1) \\rightarrow \\mathcal{Q}(-1) \\rightarrow 0,\\\\\n0 \\rightarrow 3\\mathcal{O}(-1) \\rightarrow 6\\mathcal{O} \\rightarrow \\Sym^2 \\mathcal{Q} \\rightarrow 0.\n\\end{gather*}\nWe combine them and get\n\\begin{gather}\n\\label{inclusion2-36}\n0 \\rightarrow \\mathcal{O}(-2) \\rightarrow \\mathcal{O}^{\\oplus 6} \\rightarrow \\Lambda \\rightarrow 0,\n\\\\\n\\label{Lambda2-36}\n0 \\rightarrow \\mathcal{Q}(-1) \\rightarrow \\Lambda \\rightarrow \\Sym^2 \\mathcal{Q}\\rightarrow 0,\n\\end{gather}\nwhere the rank $5$ bundle $\\Lambda$ is homogeneous, not completely reducible and globally generated, and its space of global sections coincides with $H^0(\\mathbb{P}^2, \\Sym^2 \\mathcal{Q}_{\\mathbb{P}^2}) \\cong \\Sym^2 V_3$. Adding $\\mathcal{O} \\rightarrow \\mathcal{O}$ to \\eqref{inclusion2-36} we get the desired description for $\\mathbb{P}(\\mathcal{O}(-2) \\oplus \\mathcal{O})$.\n\n\n\\hypertarget{Fano3--1}{\\subsection*{Fano 3--1}}\n\\subsubsection*{Mori-Mukai} Double cover of $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1$ with branch locus a divisor of degree $(2,2,2)$.\n\\subsubsection*{Our description} $\\mathscr{Z}(K(0,0,0,1) \\oplus \\mathcal{O}(0,0,0,2)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^8$, where the bundle $K$ is a uniquely defined extension in $\\Ext^2(\\mathcal{O}(0,0,1)^{\\oplus 4},\\mathcal{O}(1,-1,-1))$ on $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1$ fitting into the chain of extensions \\eqref{K3-1} below.\n\n\\subsubsection*{Identification}\nBy Lemma \\ref{lem:doublecovers}, our Fano variety is the zero locus of $\\mathcal{O}(2)$ on the projective bundle $\\mathbb{P}_{\\mathbb{P}^1\\times\\mathbb{P}^1\\times \\mathbb{P}^1}(\\mathcal{O}(-1,-1,-1) \\oplus \\mathcal{O})$. As it turns out, the latter projective bundle can be obtained as the zero locus of $K(0,0,0,1)$ over $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^8$, being $K \\in \\Ext^2(\\mathcal{O}(0,0,1)^{\\oplus 4},\\mathcal{O}(1,-1,-1))$ a uniquely defined extension on $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1$ fitting into \\eqref{K3-1} below. \n\nTo see it, we can argue as in Lemma \\ref{projBundle1-12} or Lemma \\ref{projBundle2-2}: we combine the (pull back of the) three (possibly twisted) Euler sequences\n\\begin{gather*}\n0 \\rightarrow \\mathcal{O}(-1,-1,-1) \\rightarrow \\mathcal{O}(0,-1,-1)^{\\oplus 2} \\rightarrow \\mathcal{O}(1,-1,-1) \\rightarrow 0,\\\\\n0 \\rightarrow \\mathcal{O}(0,-1,-1)^{\\oplus 2} \\rightarrow \\mathcal{O}(0,0,-1)^{\\oplus 4} \\rightarrow \\mathcal{O}(0,1,-1)^{\\oplus 2} \\rightarrow 0,\\\\\n0 \\rightarrow \\mathcal{O}(0,0,-1)^{\\oplus 4} \\rightarrow \\mathcal{O}^{\\oplus 8} \\rightarrow \\mathcal{O}(0,0,1)^{\\oplus 4} \\rightarrow 0.\n\\end{gather*}\nWe get\n\\begin{equation}\n\\label{Inclusion3-1}\n0 \\rightarrow \\mathcal{O}(-1,-1,-1) \\rightarrow \\mathcal{O}^{\\oplus 8} \\rightarrow K \\rightarrow 0,\n\\end{equation}\nwhere the rank $7$ bundle $K$, fitting into the chain of extension \\eqref{K3-1}, is homogeneous, not completely reducible and globally generated, and its space of global sections coincides with $H^0(\\mathbb{P}^1\\times\\mathbb{P}^1 \\times \\mathbb{P}^1, \n\\mathcal{O}(0,0,1)^{\\oplus 4}) \\cong (V_2)^{\\oplus 4}$. \n\\begin{equation}\n\\begin{gathered}\n\\label{K3-1}\n0 \\rightarrow \\mathcal{O}(1,-1,-1) \n\\rightarrow \\Lambda \\rightarrow \\mathcal{O}(0,1,-1)^{\\oplus 2} \\rightarrow 0,\n\\\\\n0 \\rightarrow \\Lambda \\rightarrow K \\rightarrow \\mathcal{O}(0,0,1)^{\\oplus 4} \\rightarrow 0.\n\\end{gathered}\n\\end{equation}\n\nAdding $\\mathcal{O} \\rightarrow \\mathcal{O}$ to \\eqref{Inclusion3-1} and from the previous considerations we get the conclusion.\n\n\n\\hypertarget{Fano3--2}{\\subsection*{Fano 3--2}}\n\\subsubsection*{Mori-Mukai} A divisor from $|\\mathcal{O}(2) \\otimes \\pi^*\\mathcal{O}(0,1)|$ on the projective bundle $\\pi:\\mathbb{P}(\\mathcal{O}(-1,-1) \\oplus \\mathcal{O}^{\\oplus 2} ) \\rightarrow \\mathbb{P}^1 \\times \\mathbb{P}^1$ such that $X \\cap Y$ is irreducible, where $X$ is the Fano itself and $Y \\in |\\mathcal{O}(1)|$.\n\n\\subsubsection*{Our description} $\\mathscr{Z}(\\Lambda(0,0,1) \\oplus \\mathcal{O}(0,1,2)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^5$, being $\\Lambda \\in \\Ext^1(\\mathcal{O}(0,1)^{\\oplus 2}, \\mathcal{O}(1, -1))$ a uniquely defined extension on $\\mathbb{P}^1 \\times \\mathbb{P}^1$ fitting into \\eqref{Lambda3-2} below.\n\n\n\\subsubsection*{Identification} \nWe need to find $\\mathbb{P}(\\mathcal{O}(-1,-1) \\oplus \\mathcal{O}^{\\oplus 2} )$ over $\\mathbb{P}^1 \\times \\mathbb{P}^1$. To do that, we argue as in Lemma \\ref{projBundle1-12} or Lemma \\ref{projBundle2-2}: we combine the two Euler exact sequences\n\\begin{gather*}\n0 \\rightarrow \\mathcal{O}(-1,-1) \\rightarrow \\mathcal{O}(0,-1)^{\\oplus 2} \\rightarrow \\mathcal{O}(1,-1) \\rightarrow 0,\\\\\n0 \\rightarrow \\mathcal{O}(0,-1)^{\\oplus 2} \\rightarrow \\mathcal{O}^{\\oplus 4} \\rightarrow \\mathcal{O}(0,1)^{\\oplus 2} \\rightarrow 0,\n\\end{gather*}\nand get\n\n\n\n\\begin{gather}\n\\label{inclusion3-2}\n0 \\rightarrow \\mathcal{O}(-1,-1) \\rightarrow \\mathcal{O}^{\\oplus 4} \\rightarrow \\Lambda \\rightarrow 0,\n\\\\\n\\label{Lambda3-2}\n0 \\rightarrow \\mathcal{O}(1,-1) \\rightarrow \\Lambda \\rightarrow \\mathcal{O}(0,1)^{\\oplus 2} \\rightarrow 0.\n\\end{gather}\nwhere the rank $3$ bundle $\\Lambda$ is homogeneous, not completely reducible and globally generated, and its space of global sections coincides with $H^0(\\mathbb{P}^1 \\times \\mathbb{P}^1, \\mathcal{O}(0,1)^{\\oplus 2}) \\cong V_2^{\\oplus 2}$. Adding $\\mathcal{O}^{\\oplus 2} \\rightarrow \\mathcal{O}^{\\oplus 2}$ to \\eqref{inclusion3-2} we get the desired description for $\\mathbb{P}(\\mathcal{O}(-1,-1) \\oplus \\mathcal{O}^{\\oplus 2})$ and the conclusion.\n\n\n\n\\hypertarget{Fano3--4}{\\subsection*{Fano 3--4}}\n\\subsubsection*{Mori-Mukai} Blow up of \\hyperlink{Fano2--18}{2--18} in a smooth fiber of the composition of the double cover projection to $\\mathbb{P}^1 \\times \\mathbb{P}^2$ with the projection to $\\mathbb{P}^2$.\n\\subsubsection*{Our description}$\\mathscr{Z}(\\Lambda(0,0,1,0) \\oplus \\mathcal{O}(0,0,2,0) \\oplus \\mathcal{O}(0,1,0,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^6 \\times \\mathbb{P}^1$, where the bundle $\\Lambda \\in \\Ext^1(\\mathcal{Q}_{\\mathbb{P}^2}^{\\oplus 2},\\mathcal{O}(1,-1))$ is a uniquely defined extension on $\\mathbb{P}^1 \\times \\mathbb{P}^2$ fitting into sequence \\eqref{Lambda2-18}.\n\\subsubsection*{Identification} The first two bundles define $Y \\times \\mathbb{P}^1$, being $Y \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^6$ the Fano \\hyperlink{Fano2--18}{2--18}. The curve on $Y$ we need to blow up is a complete intersection of two $(0,1,0)$ divisors, which cut in $Y$ the preimage of a $\\mathbb{P}^1$-fiber of the projection $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\rightarrow \\mathbb{P}^2$. We therefore conclude by Lemma \\ref{lem:blowup}.\n\n\n\n\n\n\\hypertarget{Fano3--5}{\\subsection*{Fano 3--5}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{P}^1 \\times \\mathbb{P}^2$ in a curve $C$ of degree $(5,2)$ such that $C \\hookrightarrow \\mathbb{P}^1 \\times \\mathbb{P}^2 \\rightarrow \\mathbb{P}^2$ is an embedding.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\Lambda(0,0,1) \\oplus \\mathcal{O}(0,1,1)^{\\oplus 2}) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^7$, with $\\Lambda \\in \\Ext^1_{\\mathbb{P}^1 \\times \\mathbb{P}^2}(\\mathcal{Q}_{\\mathbb{P}^2}^{\\oplus 2}, \\mathcal{O}(1,-1))$ fitting into \\eqref{Lambda2-18}.\n\n\\subsubsection*{Identification}\nBy Lemma \\ref{lem:blowDegeneracyLocus} and Lemma \\ref{lem:expectedRes} below, our Fano is the zero locus of $\\pi^* (\\mathcal{O}(0,1)^{\\oplus 2}) \\otimes \\mathcal{O}(1)$ over the projective bundle $\\pi:\\mathbb{P}(\\mathcal{O}(-1,-1) \\oplus \\mathcal{O}^{\\oplus 2}) \\rightarrow \\mathbb{P}^1 \\times \\mathbb{P}^2$. We thus need to find the latter projective bundle as the zero locus of a suitable vector bundle.\n\nA straightforward modification of the argument used for \\hyperlink{Fano2--18}{2--18} provides the desired description: adding $\\mathcal{O}^{\\oplus 2} \\rightarrow \\mathcal{O}^{\\oplus 2}$ to \\eqref{inclusion2-18} we get\n\\[\n0 \\rightarrow \\mathcal{O}(-1,-1) \\oplus \\mathcal{O}^{\\oplus 2} \\rightarrow \\mathcal{O}^{\\oplus 8} \\rightarrow \\Lambda \\rightarrow 0,\n\\]\nwhere $\\Lambda$ fits into \\eqref{Lambda2-18}. The conclusion follows as soon as we have proved the following lemma.\n\n\\begin{lemma}\\label{lem:expectedRes}\nThe ideal sheaf of a general rational curve $C$ of bidegree $(5,2)$ in $\\mathbb{P}^1 \\times \\mathbb{P}^2$ admits a locally free resolution of the form\n\\begin{equation}\n\\label{shaperesolution}\n0 \\rightarrow\n\\mathcal{O}(-1,-4)^{\\oplus 2} \\rightarrow\n\\mathcal{O}(0,-2) \\oplus \\mathcal{O}(-1,-3)^{\\oplus 2} \\rightarrow\n\\mathcal{I}_C \\rightarrow 0\n\\end{equation}\nand, conversely, a general $3 \\times 2$ matrix as above yields a presentation for the ideal sheaf of a general curve $C$.\n\\begin{proof}\nThe aim is to show, on the one hand, that the above resolution is the simplest (in terms of Betti numbers) such a curve is expected to have. On the other hand, if we manage to show that a curve having that resolution exists, a semicontinuity argument yields that a general curve shares the same behaviour.\n\nThe first task requires a bit of commutative algebra, which we specialise to our setting $\\mathbb{P}:=\\mathbb{P}^1 \\times \\mathbb{P}^2$. Let $R:=\\oplus_{(a,b) \\in \\mathbb{Z}^2}H^0(\\mathbb{P},\\mathcal{O}(a,b))$ be the Cox ring of $\\mathbb{P}$. If $I_C$ denotes the ideal of $C$, which can be seens as a finitely generated $R$-module, we have a multigraded minimal free resolution\n\\[\n0 \\rightarrow F_r \\rightarrow \\dotso \\rightarrow F_0 \\rightarrow I_C \\rightarrow 0,\n\\]\nwhere the $F_i$ are finitely generated free modules $F_i = \\oplus_{(a,b) \\in \\mathbb{Z}^2}R(-a,-b)^{\\oplus \\beta_{i,(a,b)}}$, being $\\beta_{i,(a,b)}$ the so-called multigraded Betti numbers, which are independent of the chosen resolution.\n\nThe so-called multigraded Hilbert series of $I_C$ is the formal Laurent series\n\\[\nH_{I_C}:=\\sum_{(a,b) \\in \\mathbb{Z}^2} \\dim_{\\mathbb{C}}(I_C)_{(a,b)}\\cdot s^a t^b,\n\\]\nwhich is well-known to encode the Betti numbers $\\beta_{i,(a,b)}$ in the following way: it factors as\n\\[\nH_{I_C}=\\frac{\n\\sum_{(a,b) \\in \\mathbb{Z}^2}\\left( \\sum_{i=0}^r (-1)^{i} \\beta_{i,(a,b)}\\right) \\cdot s^a t^b\n}{(1-s)^2(1-t)^3}.\n\\]\n\nBy Riemann--Roch we can compute $H^0(C,\\mathcal{O}_{C}(a,b))$ for any $(a,b) \\in \\mathbb{Z}^2$; if we assume that $C$ has maximal rank, i.e., that $H^0(\\mathbb{P},\\mathcal{O}_{\\mathbb{P}}(a,b)) \\rightarrow H^0(C,\\mathcal{O}_{C}(a,b))$ has maximal rank for all $(a,b) \\in \\mathbb{Z}^2$, then we explicitly have $\\dim_{\\mathbb{C}}(I_C)_{(a,b)}$ and $H_{I_C}$. Straightforward computations then show that the numerator of $H_{S\/I_C}$ is $t^2 + 2st^3 -2st^4$; thus, the expected resolution of $I_C$ has the shape \\eqref{shaperesolution}.\n\nTo conclude, it suffices to show the existence of a curve with the right genus and degree having the desired resolution. This task can be rather difficult, depending on the given invariants: on $\\mathbb{P}^1 \\times \\mathbb{P}^2$, different approaches can be adopted, such as liaison theory or the construction of the Hartshorne--Rao module of the curve, see, e.g., \\cite{KeneshlouTanturri1,KeneshlouTanturri2}. Our situation, however, is favourable, as the minors of a general matrix\n\\[\n\\mathcal{O}(-1,-4)^{\\oplus 2} \\rightarrow\n\\mathcal{O}(0,-2) \\oplus \\mathcal{O}(-1,-3)^{\\oplus 2}\n\\]\ngenerate the ideal of a smooth curve of maximal rank with the desired invariants. This can be checked via any computer algebra software like \\cite{M2}.\n\\end{proof}\n\\end{lemma}\n\n\nIf we consider the normal sequence for $Y=\\mathscr{Z}(\\mathcal{F}) \\subset \\mathbb{P}:=\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^7$, a few cohomology computations via the Koszul complex as described in Section \\ref{computeinvariants} provide that $h^0(T_{\\mathbb{P}}|_Y)=74, h^0(\\mathcal{F}|_Y)=79$ and the higher cohomology groups vanish. In \\cite[Corollary 8.8]{pcs} it is shown that the family of Fano \\hyperlink{Fano3--5}{3--5} has a unique member with infinite automorphism group. This means that a general model $Y$ admits a $(79-74=5)$-dimensional family of deformations, which is the dimension of the moduli of Fano \\hyperlink{Fano3--5}{3--5}, hence $Y$ is general in moduli.\n\n\\hypertarget{Fano3--6}{\\subsection*{Fano 3--6}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{P}^3$ in the disjoint union of a line and an elliptic curve of degree 4.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,0,2) \\oplus \\mathcal{O}(0,1,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^3$.\n\\subsubsection*{Identification}\nA quartic elliptic curve is a given by a complete intersections of two quadrics in $\\mathbb{P}^3$. It then suffices to apply twice Lemma \\ref{lem:blowup}.\n\n\n\\hypertarget{Fano3--8}{\\subsection*{Fano 3--8}}\n\\subsubsection*{Mori-Mukai} Divisor from the linear system $|(\\alpha \\circ \\pi)^* (\\mathcal{O}_{\\mathbb{P}^2}(1)) \\boxtimes \\mathcal{O}_{\\mathbb{P}^2}(2)|$ on $\\Bl_p \\mathbb{P}^2 \\times \\mathbb{P}^2$, where $\\pi: \\Bl_p \\mathbb{P}^2 \\times \\mathbb{P}^2 \\rightarrow \\Bl_p \\mathbb{P}^2$ is the first projection and $\\alpha: \\Bl_p \\mathbb{P}^2 \\rightarrow \\mathbb{P}^2$ is the blow up map. \n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(0,1,2) \\oplus \\mathcal{O}(1,1,0)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^2$.\n\n\\subsubsection*{Identification} See Lemma \\ref{lem:blowup}.\n\n\\hypertarget{Fano3--9}{\\subsection*{Fano 3--9}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{P}_{\\mathbb{P}^2}(\\mathcal{O} \\oplus \\mathcal{O}(-2))$ in a quartic curve on $\\mathbb{P}^2$.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\Lambda(0,1,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^6}(0,0,1) \\oplus K(0,0,1)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^6 \\times \\mathbb{P}^{20}$, where the bundle $\\Lambda\\in \\Ext^1(\\Sym^2 \\mathcal{Q},\\mathcal{Q}(-1))$ is a uniquely defined extension on $\\mathbb{P}^2$ fitting into sequence \\eqref{Lambda2-36} and $K \\in \\Ext^3(\\Sym^4 \\mathcal{Q},\\mathcal{Q}(-3))$ is a uniquely defined extension on $\\mathbb{P}^2$ fitting into the chain of extensions \\eqref{Kappa3--9}.\n\n\\subsubsection*{Identification}\nWe need to blow up \\hyperlink{Fano2--36}{2--36} in a quartic curve $C$ on the base $\\mathbb{P}^2$. The first bundle defines $Y:=$ \\hyperlink{Fano2--36}{2--36} inside $\\mathbb{P}^2\\times \\mathbb{P}^6$; since $C$ is the zero locus of a map\n\\[\n\\mathcal{O}(0,-1) \\oplus \\mathcal{O}(-4,0) \\rightarrow \\mathcal{O}\n\\]\non $Y$, by Lemma \\ref{lem:blowDegeneracyLocus} our Fano will be the zero locus of $\\mathcal{O}(1)$ over $\\mathbb{P}_Y(\\mathcal{O}(0,-1) \\oplus \\mathcal{O}(-4,0))$.\n\nFor the first bundle $\\mathcal{O}(0,-1)$, we have the standard (pulled back) Euler sequence\n\\begin{equation}\n\\label{of0-1}\n0 \\rightarrow\n\\mathcal{O}(0,-1) \\rightarrow\n\\mathcal{O}^{\\oplus 7} \\rightarrow\n\\mathcal{Q}_{\\mathbb{P}^6} \\rightarrow\n0;\n\\end{equation}\nthe second bundle $\\mathcal{O}(-4,0)$ requires a cumbersome though straightforward merging of the following (dualised) short exact sequences on $\\mathbb{P}^2$ given by Remark \\ref{rem:principalParts}:\n\\begin{gather*}\n0 \\rightarrow\n\\mathcal{O}(-4) \\rightarrow\n\\mathcal{O}(-3)^{\\oplus 3} \\rightarrow\n\\mathcal{Q}(-3) \\rightarrow\n0\\\\\n0 \\rightarrow\n\\mathcal{O}(-3)^{\\oplus 3} \\rightarrow\n\\mathcal{O}(-2)^{\\oplus 6} \\rightarrow\n\\Sym^2\\mathcal{Q}(-2) \\rightarrow\n0\\\\\n0 \\rightarrow\n\\mathcal{O}(-2)^{\\oplus 6} \\rightarrow\n\\mathcal{O}(-1)^{\\oplus 10} \\rightarrow\n\\Sym^3\\mathcal{Q}(-1) \\rightarrow\n0\\\\\n0 \\rightarrow\n\\mathcal{O}(-1)^{\\oplus 10} \\rightarrow\n\\mathcal{O}^{\\oplus 15} \\rightarrow\n\\Sym^4\\mathcal{Q} \\rightarrow\n0.\n\\end{gather*}\nArranging them repeatedly as in Lemma \\ref{projBundle1-12} or Lemma \\ref{projBundle2-2}, we get to a uniquely defined homogeneous rank $14$ vector bundle $K$ on $\\mathbb{P}^2$ which fits into\n\\begin{equation}\n\\label{of-40}\n0 \\rightarrow\n\\mathcal{O}(-4) \\rightarrow\n\\mathcal{O}^{\\oplus 15} \\rightarrow\nK \\rightarrow\n0\n\\end{equation}\nand into the following chain of extensions\n\\begin{equation}\n\\begin{gathered}\n\\label{Kappa3--9}\n0 \\rightarrow\n\\mathcal{Q}(-3) \\rightarrow\nK_1 \\rightarrow\n\\Sym^2\\mathcal{Q}(-2) \\rightarrow\n0\\\\\n0 \\rightarrow\nK_1 \\rightarrow\nK_2 \\rightarrow\n\\Sym^3\\mathcal{Q}(-1) \\rightarrow\n0\\\\\n0 \\rightarrow\nK_2 \\rightarrow\nK\\rightarrow\n\\Sym^4\\mathcal{Q} \\rightarrow\n0.\n\\end{gathered}\n\\end{equation}\nOne can directly check using \\eqref{of-40} and \\eqref{Kappa3--9} that $H^0(K) \\cong \\Sym^4 V_3$ and $H^1(K) \\cong V_3$.\nThe conclusion follows by considering the direct sum of \\eqref{of-40} and \\eqref{of0-1}.\n\n\n\n\\hypertarget{Fano3--10}{\\subsection*{Fano 3--10}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{Q}_3$ in the disjoint union of 2 conics.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,0,1) \\oplus \\mathcal{O}(0,1,1) \\oplus \\mathcal{O}(0,0,2)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^4$.\n\\subsubsection*{Identification} It suffices to apply twice Lemma \\ref{lem:blowup}.\n\n\n\\hypertarget{Fano3--11}{\\subsection*{Fano 3--11}}\n\n\\subsubsection*{Mori-Mukai} Blow up of \\hyperlink{Fano2--35}{2--35} in an elliptic curve which is the intersection of two divisors from $|-\\frac{1}{2}K|.$\n\\subsubsection*{Our description} \n$\\mathscr{Z}(\\mathcal{O}(1,1,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(0,0,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^3$.\n\\subsubsection*{Identification} We recall first that \\hyperlink{Fano2--35}{2--35} is the blow up of $\\mathbb{P}^3$ at a point, which we have already identified as $\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^2}(0,1)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^3$. As such, its anticanonical class is $\\mathcal{O}(2,2)$ by adjunction. It then suffices to apply Lemma \\ref{lem:blowup}.\n\n\n\\hypertarget{Fano3--12}{\\subsection*{Fano 3--12}}\n\\subsubsection*{Mori-Mukai} \t\nBlow up of $\\mathbb{P}^3$ in the disjoint union of a line and a twisted cubic.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(0,1,1)\\oplus \\mathcal{O}(0,1,1) \\oplus \\mathcal{O} (1,0,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^3.$\n\n\\subsubsection*{Identification} The variety $\\mathscr{Z}(\\mathcal{O}(1,1)\\oplus \\mathcal{O}(1,1) ) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^3$ is the Fano 3-fold 2--27, the blow up of $\\mathbb{P}^3$ in a twisted cubic. The result then follows by Lemma \\ref{lem:blowup}, with the two extra $(0,1)$ divisors cutting a line in space which by construction is disjoint from the twisted cubic. To make this explicit, take coordinates $[z_0,z_1], \\ [y_0, y_1, y_2], \\ [x_0,\\ldots, x_3]$. The divisor of degree $(1,0,1)$ is therefore given by an expression of type $\\sum z_i f_i(x_i)$. Say for simplicity $z_0 x_0 +z_1 x_3$. The line $L$ in $\\mathbb{P}^3$ which we are blowing up is therefore given by $x_0=x_3=0$. On the other hand the two divisors of degree $(0,1,1)$ define the twisted cubic as follows: they are given by the solutions of, e.g., \\[ \n {\\begin{pmatrix}\n x_0 & x_1 & x_2 \\\\\n x_1 & x_2 & x_3 \\\\\n \\end{pmatrix} }\n {\\begin{pmatrix} y_0 \\\\\n y_1 \\\\\n y_2\n \\end{pmatrix}}=0.\n \\]\nIn particular this locus is trivially identified with the blow up of $\\mathbb{P}^3$ where the matrix drops rank, that is $ {\\rank \\begin{pmatrix}\n x_0 & x_1 & x_2 \\\\\n x_1 & x_2 & x_3 \\\\\n \\end{pmatrix} } <2$. The latter are the equations of the twisted cubic in $\\mathbb{P}^3$, which we can easily check to be disjoint from the line $L$.\n\n\\hypertarget{Fano3--14}{\\subsection*{Fano 3--14}}\n\\subsubsection*{Mori-Mukai} \t\nBlow up of $\\mathbb{P}^3$ in the disjoint union of a plane cubic curve and a point outside the plane.\n\\subsubsection*{Our description} $\\mathscr{Z}( \\Lambda(0,1,0) \\oplus \\mathcal{O}(1,1,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(1,0,0)) \\subset \\mathbb{P}^3 \\times \\mathbb{P}^{10} \\times \\mathbb{P}^2$, where the bundle $\\Lambda \\in \\Ext^1(\\Sym^2 \\mathcal{Q},\\mathcal{Q}(-1))$ is a uniquely defined extension on $\\mathbb{P}^3$ fitting into sequence \\eqref{Lambda1--12} above.\n\n\n\n\n\\subsubsection*{Identification} The first two bundles on $\\mathbb{P}^3 \\times \\mathbb{P}^{10}$ determine \\hyperlink{Fano2--28}{2--28}, i.e., the blow up of $\\mathbb{P}^3$ in a plane cubic curve. To blow it up in a point, we can apply Lemma \\ref{lem:blow} for the base $\\mathbb{P}^3$, adding a $\\mathbb{P}^2$ factor and the corresponding bundle. The extra point will in general be outside the plane.\n\n\\hypertarget{Fano3--15}{\\subsection*{Fano 3--15}}\n\\subsubsection*{Mori-Mukai} \t\nBlow up of $\\mathbb{Q}_3$ in the disjoint union of a line and a conic.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,0,1) \\oplus \\mathcal{O}(0,1,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(0,0,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^4$.\n\n\\subsubsection*{Identification} By Lemma \\ref{lem:blow} the zero locus of the last two bundles on $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^4$ gives us $\\mathbb{P}^1 \\times \\Bl_{\\mathbb{P}^1}\\mathbb{Q}_3$. We still have to cut with a section of $\\mathcal{O}(1,0,1)$. By Lemma \\ref{lem:blowup} this is the blow up of $\\Bl_{\\mathbb{P}^1} \\mathbb{Q}_3$ in the locus cut by two linear sections, which is in general disjoint from the $\\mathbb{P}^1$. The result follows.\n\n\\hypertarget{Fano3--16}{\\subsection*{Fano 3--16}} \n\\subsubsection*{Mori-Mukai} Blow up of \\hyperlink{Fano2--35}{2--35} in the proper transform of a twisted cubic containing the centre of the blow up.\n\\subsubsection*{Our description}\n$\\mathscr{Z}(\\mathcal{O}(0,1,1) \\oplus \\mathcal{O}(1,0,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2_1}(0,1,0)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^3 \\times \\mathbb{P}^2$.\n\n\\subsubsection*{Identification} We first fix the system of coordinates\n$\\mathbb{P}^2_{[y_0\\ldots y_2]} \\times \\mathbb{P}^3_{[x_0\\ldots x_3]} \\times \\mathbb{P}^2_{[w_0\\ldots w_2]}.$ As a first step we use Lemma \\ref{lem:blow} to identify $\\mathcal{Q}_{\\mathbb{P}^2} (0,1) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^3 $ as \\hyperlink{Fano2--35}{2--35}, i.e., $\\Bl_p \\mathbb{P}^3$. The two remaining divisors are, on $\\mathbb{P}^2 \\times \\mathbb{P}^3$, of degree $(0,1)$ and $(1,0)$ and are both trivially identified with linear forms on $\\mathbb{P}^3$, but with a distinction. Without loss of generality, assume that $p$ is the point $[1,0,0,0]$. We have $(f \\in |\\mathcal{O}(1,0)|) \\in \\mathrm{Ann}(p)$, while $(g \\in |\\mathcal{O}(0,1)|)$ gives a non-zero element of $V_4^{\\vee}\/\\mathrm{Ann}(p)$. In other words, $f=f(x_1,x_2,x_3)$ does not contain the coordinate $x_0$, while the converse holds for $g$. Both the divisors were twisted by $\\mathcal{O}_{\\mathbb{P}^2}(1)$, giving rise to two divisors of degree $(1,1)$ on $\\Bl_p\\mathbb{P}^3 \\times \\mathbb{P}^2_{[w_0\\ldots w_2]}$. As in \\hyperlink{Fano3--12}{3--12}, these lead to the blow up of $\\Bl_p\\mathbb{P}^3$ in a twisted cubic, that (since $f \\in \\mathrm{Ann}(p)$) passes through the point $p \\in \\mathbb{P}^3$. The result follows.\n\n\n\\hypertarget{Fano3--18}{\\subsection*{Fano 3--18}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{P}^3$ in the disjoint union of a line and a conic.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_2(0;0,0) \\oplus \\mathcal{O}(0;1,1) \\oplus \\mathcal{O}(1;0,1)) \\subset \\mathbb{P}^1 \\times \\Fl(1,2,5)$.\n\n\\subsubsection*{Identification}\nThis Fano can be evidently identified with the blow up of \\hyperlink{Fano2--30}{2--30} in a line disjoint from the conic. Recall that we described \\hyperlink{Fano2--30}{2--30} as $(\\mathscr{Z}(\\mathcal{Q}_2 \\oplus \\mathcal{O}(1,1)) \\subset \\Fl(1,2,5)) \\cong \\mathscr{Z}(\\mathcal{O}(1)) \\subset \\mathbb{P}_{\\mathbb{P}^3}(\\mathcal{O}(-1)\\oplus \\mathcal{O})$. The result then follows from Lemma \\ref{lem:blowup}, since two divisors of degree $(0,1)$ cut a line in the base $\\mathbb{P}^3$.\n\nWe can write an alternative description for this Fano, based on the alternative description already given for \\hyperlink{Fano2--30}{2--30}. Using Lemma \\ref{lem:blowup} the Fano \\hyperlink{Fano3--18}{3--18} will be\n\n\\[\n\\mathscr{Z}(\\mathcal{O}(1,1,0) \\oplus \\mathcal{O}(0,1,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^3}(0,0,1)) \\subset \\mathbb{P}^1\\times \\mathbb{P}^3 \\times \\mathbb{P}^4.\n\\]\n\n\\hypertarget{Fano3--19}{\\subsection*{Fano 3--19}}\n\\subsubsection*{Mori-Mukai}Blow up of $\\mathbb{Q}_3$ in two non-collinear points.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^2}(0,1) \\oplus \\mathcal{O}(0,2)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^4$.\n\n\\subsubsection*{Identification}\nBy Lemma \\ref{lem:blow}, the first divisor yields the blow up of $\\mathbb{P}^4$ along a line. The second divisor is identified with a general quadric in $\\mathbb{P}^4$, hence it cuts out a quadric hypersurface in $\\mathbb{P}^4$ blown up along two points. The general quadric does not contain the line, so the blown up points are in general non-collinear.\n\n\\hypertarget{Fano3--20}{\\subsection*{Fano 3--20}} \n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{Q}_3$ in the disjoint union of two lines.\n\\subsubsection*{Our description}\n$\\mathscr{Z}(\\mathcal{O}(1,0,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2_1}(0,1,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^2_2}(0,1,0) ) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^5 \\times \\mathbb{P}^2$.\n\n\\subsubsection*{Identification} We remark that, by Lemma \\ref{lem:blow}, another model for \\hyperlink{Fano2--31}{2--31} (the blow up of $\\mathbb{Q}_3$ in one line) is given by $\\mathscr{Z}(\\mathcal{O}(1,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2} (0,1)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^5$. Our model for \\hyperlink{Fano3--20}{3--20} is just the iteration of the blow up process, where the second and the third bundles give the blow up of $\\mathbb{P}^4$ along two disjoint lines $L_1, L_2$ and the first bundle gives a quadric which contains both the lines. Notice that in fact a section $\\sum_k f_{1,k} f_{2,k}$ of the bundle $\\mathcal{O}(1,0,1)$ identifies a quadric in $\\mathbb{P}^4$ and $f_{i,k} \\in \\Ann(L_i)$ for $i=1,2$ (see also the arguments used for \\hyperlink{Fano2--19}{2--19} and \\hyperlink{Fano3--16}{3--16}).\n\n\\hypertarget{Fano3--21}{\\subsection*{Fano 3--21}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{P}^1 \\times \\mathbb{P}^2$ in a curve of degree $(2,1)$.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(0,1,1) \\oplus \\Lambda(0,0,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^6$, being $\\Lambda \\in \\Ext^1(\\mathcal{Q}_{\\mathbb{P}^2}^{\\oplus 2},\\mathcal{O}(1,-1))$ a uniquely defined extension on $\\mathbb{P}^1 \\times \\mathbb{P}^2$ fitting into sequence \\eqref{Lambda2-18}.\n\n\\subsubsection*{Identification}\nOn $\\mathbb{P}^1 \\times \\mathbb{P}^2$, a general complete intersection of a $(0,1)$ and a $(1,2)$ divisors is a smooth curve of degree $(2,1)$. In order to blow it up, we can use Lemma \\ref{lem:blowDegeneracyLocus}, according to which our Fano will be the zero locus of $\\mathcal{O}(1) \\otimes \\pi^*(0,1)$ over the projective bundle $\\pi:\\mathbb{P}(\\mathcal{O}(-1,-1) \\oplus \\mathcal{O})\\rightarrow \\mathbb{P}^1 \\times \\mathbb{P}^2$.\n\nThe above projective bundle has already been found when dealing with \\hyperlink{Fano2--18}{2--18}: it is the zero locus of $\\Lambda(0,0,1)$ over $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^6$, with $\\Lambda$ fitting into \\eqref{Lambda2-18}.\n\n\\hypertarget{Fano3--22}{\\subsection*{Fano 3--22}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{P}^1 \\times \\mathbb{P}^2$ in a conic on $\\lbrace x \\rbrace \\times \\mathbb{P}^2 $, $\\lbrace x \\rbrace \\in \\mathbb{P}^1$.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,0,1) \\oplus \\Lambda(0,0,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^6$, being $\\Lambda \\in \\Ext^1(\\Sym^2 \\mathcal{Q},\\mathcal{Q}(-1))$ a uniquely defined extension on $\\mathbb{P}^2$ fitting into sequence \\eqref{Lambda2-36}.\n\\subsubsection*{Identification}\nWe need to blow up on $\\mathbb{P}^1 \\times \\mathbb{P}^2$ a complete intersection curve given by two divisors of degree $(1,0)$ and $(0,2)$. To do that, we use Lemma \\ref{lem:blowDegeneracyLocus}: our Fano will then be the zero locus of $\\mathcal{O}(1)$ over the projective bundle $\\mathbb{P}(\\mathcal{O}(-1,0) \\oplus \\mathcal{O}(0,-2))$.\n\nTo find the above projective bundle, we can add the standard (pulled back) Euler sequence on $\\mathbb{P}^1$ to \\eqref{inclusion2-36} and get\n\\[\n0 \\rightarrow\n\\mathcal{O}(-1,0) \\oplus \\mathcal{O}(0,-2) \\rightarrow\n\\mathcal{O}^{\\oplus 8} \\rightarrow\n\\mathcal{O}(1,0) \\oplus \\Lambda \\rightarrow\n0,\n\\]\nbeing $\\Lambda$ a uniquely defined extension on $\\mathbb{P}^2$ fitting into sequence \\eqref{Lambda2-36}. The conclusion follows.\n\n\\hypertarget{Fano3--23}{\\subsection*{Fano 3--23}}\n\\subsubsection*{Mori-Mukai} Blow up of \\hyperlink{Fano2--35}{2--35} in the proper transform of a conic containing the centre of the blow up.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^2}(0,1,0) \\oplus \\mathcal{O}(1,0,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^3}(0,0,1)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^3 \\times \\mathbb{P}^4$.\n\n\\subsubsection*{Identification} By Lemma \\ref{lem:blow} the first bundle (when seen on the first two factors) gives $X:=$ \\hyperlink{Fano2--35}{2--35}, the blow up of $\\mathbb{P}^3$ in one point $p$. We need to blow up $X$ along the proper transform of a conic $Q$ containing $p$. Note that $Q$ is cut out by a hyperplane and a quadric in $\\mathbb{P}^3$ both containing $p$, so that $Q$ is the degeneracy locus of a map $\\mathcal{O}_X(-1,-1) \\oplus \\mathcal{O}_X(-1,0) \\rightarrow \\mathcal{O}_X$ (see, e.g., the arguments used for \\hyperlink{Fano2--19}{2--19} and \\hyperlink{Fano3--16}{3--16}). Lemma \\ref{lem:blowDegeneracyLocus} yields that our Fano will be the zero locus of $\\mathcal{O}(1) \\otimes \\pi^*(\\mathcal{O}(1,0))$ over the projective bundle $\\pi:\\mathbb{P}(\\mathcal{O}(0,-1) \\oplus \\mathcal{O}) \\rightarrow \\mathbb{P}^2 \\times \\mathbb{P}^3$.\n\nSuch projective bundle can be found in $\\mathbb{P}^2 \\times \\mathbb{P}^3 \\times \\mathbb{P}^4$, as the sequence on $\\mathbb{P}^2 \\times \\mathbb{P}^3$\n\\[\n0 \\rightarrow\n\\mathcal{O}(0,-1) \\oplus \\mathcal{O} \\rightarrow\n\\mathcal{O}^{\\oplus 5} \\rightarrow\n\\mathcal{Q}_{\\mathbb{P}^3} \\rightarrow\n0\n\\]\nshows. The conclusion follows.\n\n\\hypertarget{Fano3--24}{\\subsection*{Fano 3--24}}\n\\subsubsection*{Mori-Mukai} The fiber product of 2--32 with $\\Bl_p\\mathbb{P}^2$ over $\\mathbb{P}^2$.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,1,0) \\oplus \\mathcal{O}(0,1,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^2.$\n\\subsubsection*{Identification} See \\cite[\\textsection 77]{corti}.\n\n\\hypertarget{Fano3--25}{\\subsection*{Fano 3--25}}\n\\subsubsection*{Mori-Mukai} $\\mathbb{P}_{\\mathbb{P}^1 \\times \\mathbb{P}^1}(\\mathcal{O}(0,-1) \\oplus \\mathcal{O}(-1,0))$, or the blow up of $\\mathbb{P}^3$ in two disjoint lines.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(0,1)^{\\oplus 2}) \\subset \\Fl(1,2,4)$.\n\n\\subsubsection*{Identification} We can identify $\\Fl(1,2,4)$ with $\\mathbb{P}_{\\Gr(2,4)}(\\mathcal{U})$. Let $Z:= \\mathbb{P}^1 \\times \\mathbb{P}^1$. The two $(0,1)$ sections give us $\\mathbb{P}_Z(\\mathcal{U}|_Z)$. By \\cite[Theorem 1.4]{ottaviani} the restriction of $\\mathcal{U}$ to $Z$ coincides with the direct sum of $\\mathcal{O}(0,-1) \\oplus \\mathcal{O}(-1,0)$. The result follows. \n\n\nAn alternative description is $\\mathscr{Z}(\\mathcal{O}(1,1,0) \\oplus \\mathcal{O}(1,0,1)) \\subset \\mathbb{P}^3 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1$, by simply apply twice Lemma \\ref{lem:blowup}.\n\n\\hypertarget{Fano3--26}{\\subsection*{Fano 3--26}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{P}^3$ in the disjoint union of a point and a line.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,0,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(0,0,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^3$.\n\n\\subsubsection*{Identification} The bundle $\\mathcal{Q}_{\\mathbb{P}^2} (0,1)$ on $ \\mathbb{P}^2 \\times \\mathbb{P}^3$ gives the Fano \\hyperlink{Fano2--35}{2--35} by Lemma \\ref{lem:blow}. Two extra sections of $\\mathcal{O}(0,1)$ on this space cut a line that does not intersect the exceptional divisor (equivalently, a line in $\\mathbb{P}^3$ that does not pass through the blown up point). The identification therefore follows by Lemma \\ref{lem:blowup}.\n\n\\hypertarget{Fano3--28}{\\subsection*{Fano 3--28}}\n\\subsubsection*{Mori-Mukai} $\\mathbb{P}^1 \\times \\Bl_p \\mathbb{P}^2$.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,0,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^2$.\n\\subsubsection*{Identification} See Lemma \\ref{lem:blowup}.\n\n\n\\hypertarget{Fano3--29}{\\subsection*{Fano 3--29}}\n\\subsubsection*{Mori-Mukai} Blow up of \\hyperlink{Fano2--35}{2--35} in a line on the exceptional divisor.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^2}(1,0,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^3}(0,0,1) \\oplus \\Lambda(0,0,1) \\oplus \\mathcal{O}(0,-1,1)) \\subset \\mathbb{P}^3 \\times \\mathbb{P}^2 \\times \\mathbb{P}^9$, being $\\Lambda \\in \\Ext^1(\\Sym^2 \\mathcal{Q},\\mathcal{Q}(-1))$ a uniquely defined extension on $\\mathbb{P}^2$ fitting into sequence \\eqref{Lambda2-36}.\n\\subsubsection*{Identification} By Lemma \\ref{lem:blow} the first bundle gives, on the first two factors, the blow up $Y$ of $\\mathbb{P}^3$ along a point. We then need to blow up a line in the exceptional divisor. By \\cite[Corollary 9.12]{EisenbudHarris3264}, the exceptional divisor is a $(1,-1)$ divisor in $Y$; in order to cut out a line on it, we have to intersect it with the strict transform of a hyperplane in $\\mathbb{P}^3$ passing through the point, which is a $(0,1)$ divisor (see, e.g., the argument used for \\hyperlink{Fano3--16}{3--16}).\n\nSummarising, we need to blow $Y$ up along the intersection of the two aforementioned divisors. By Lemma \\ref{lem:blowDegeneracyLocus}, this yields that our Fano variety is the zero locus of $\\pi^* \\mathcal{O}(0,-1) \\otimes \\mathcal{O}(1)$ over $\\pi:\\mathbb{P}(\\mathcal{O}(-1,0) \\oplus \\mathcal{O}(0,-2)) \\rightarrow Y$. \n\nTo express the above projective bundle, we can add the standard (pulled back) Euler sequence on $\\mathbb{P}^3$ to \\eqref{inclusion2-36} and get\n\\[\n0 \\rightarrow\n\\mathcal{O}(-1,0) \\oplus \\mathcal{O}(0,-2) \\rightarrow\n\\mathcal{O}^{\\oplus 10} \\rightarrow\n\\mathcal{Q}_{\\mathbb{P}^3} \\oplus \\Lambda \\rightarrow\n0,\n\\]\nbeing $\\Lambda$ a uniquely defined extension on $\\mathbb{P}^2$ fitting into sequence \\eqref{Lambda2-36}. The conclusion follows.\n\n\\begin{caveat}\n\\label{caveatBundle}\nThe above bundle $\\mathcal{O}(0,-1,1)$ has clearly no sections on $\\mathbb{P}^3 \\times \\mathbb{P}^2 \\times \\mathbb{P}^9$, so our notation seems misleading. In fact, this bundle acquires a $4$-dimensional space of global sections once it is restricted to the zero locus of the previous ones, so that the direct sum above should be taken with a pinch of salt.\n\nThis phenomenon naturally occurs when we need to consider the exceptional divisor of a blow up obtained via Lemma \\ref{lem:blow}: as already remarked, if $Y=\\Bl_{\\mathbb{P}^{n-m-1}}\\mathbb{P}^n=\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^m}(0,1))\\subset \\mathbb{P}^m \\times \\mathbb{P}^n$, then the exceptional divisor is a $(-1,1)$ divisor in $Y$. Notice that $\\mathcal{O}_{\\mathbb{P}^n \\times \\mathbb{P}^m}(-1,1)|_Y \\cong \\mathcal{O}_Y(-1,1)$ indeed has global sections.\n\\end{caveat}\n\n\n\\hypertarget{Fano3--30}{\\subsection*{Fano 3--30}}\n\\subsubsection*{Mori-Mukai} Blow up of \\hyperlink{Fano2--35}{2--35} in the proper transform of a line containing the centre of the blow up.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,0,1) \\oplus \\mathcal{Q}_2(0,1,0)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^3 \\times \\mathbb{P}^1$.\n\\subsubsection*{Identification} By Lemma \\ref{lem:blow} the first bundle (when seen on the first two factors) gives a Fano $X$ which is \\hyperlink{Fano2--35}{2--35}, the blow up of $\\mathbb{P}^3$ in one point $p$. We need to blow up $X$ along the proper transform of a line containing $p$, which is the complete intersection of two divisors of degree $(1,0)$ on $\\mathbb{P}^2 \\times \\mathbb{P}^3$ (see, e.g., the argument used for \\hyperlink{Fano3--16}{3--16}). We conclude by Lemma \\ref{lem:blowup}.\n\n\\hypertarget{Fano3--31}{\\subsection*{Fano 3--31}}\n\\subsubsection*{Mori-Mukai} Blow up of the cone over a smooth quadric in $\\mathbb{P}^3$ in the vertex, or $\\mathbb{P}_{\\mathbb{P}^1 \\times \\mathbb{P}^1}(\\mathcal{O}(-1,-1) \\oplus \\mathcal{O})$.\n\\subsubsection*{Our description} of $\\mathscr{Z}(\\mathcal{Q}_2 \\oplus \\mathcal{O}(0,2)) \\subset \\Fl(1,2,5)$.\n\n\\subsubsection*{Identification} By Corollary \\ref{cor:blowupflag} we have that $\\mathscr{Z}(\\mathcal{Q}_2) \\subset \\Fl(1,2,5)$ is isomorphic to $\\mathbb{P}_{\\mathbb{P}^3}(\\mathcal{O}(-1) \\oplus \\mathcal{O})$. The extra quadric cuts only the base $\\mathbb{P}^3$, and yields the identification. \n\nWe want to give an alternative description as \\[\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^3}(0,1) \\oplus \\mathcal{O}(2,0)) \\subset \\mathbb{P}^3 \\times \\mathbb{P}^4.\\] \nBy Lemma \\ref{lem:blow}, $\\mathcal{Q}_{\\mathbb{P}^3} (0,1)$ gives the blow up of $\\mathbb{P}^4$ at a point $p_0$, with dual coordinate $x_0$. A section of $\\mathcal{O}(2,0)$ gives a quadric in the space $\\Sym^2(V_5^{\\vee}\/\\langle x_0 \\rangle)$. This gives the equation of a cone over a smooth, degenerate quadric in $\\mathbb{P}^3_{[x_1, \\ldots, x_4]}$. The result follows.\n\n\n\\hypertarget{Fano4--2}{\\subsection*{Fano 4--2}}\n\\subsubsection*{Mori-Mukai} \nBlow up of the cone over a smooth quadric in $\\mathbb{P}^3$ in the disjoint union of the vertex and an elliptic curve on the quadric. \n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^3}(0,1,0) \\oplus \\mathcal{O}(2,0,0) \\oplus \\mathcal{O}(0,1,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^4}(0,0,1)) \\subset \\mathbb{P}^3 \\times \\mathbb{P}^4 \\times \\mathbb{P}^5$.\n\n\\subsubsection*{Identification} \nWe use the alternative description of \\hyperlink{Fano3--31}{3--31}. In fact to blow up the requested elliptic curve it suffices to blow up $Y:=$ \\hyperlink{Fano3--31}{3--31}, in its intersection with a hyperplane not passing through the vertex of the cone and a general quadric, i.e., in the intersection of a $(0,1)$ and a $(0,2)$ divisors. Lemma \\ref{lem:blowDegeneracyLocus} yields that our Fano variety is the zero locus of $\\pi^* \\mathcal{O}(0,1) \\otimes \\mathcal{O}(1)$ over $\\pi:\\mathbb{P}(\\mathcal{O}(0,-1)\\oplus \\mathcal{O}) \\rightarrow Y$. Such projective bundle can be obtained as the zero locus of the remaining bundle by considering the Euler sequence on $\\mathbb{P}^4$, which yields and embedding of $\\mathcal{O}(0,-1)\\oplus \\mathcal{O}$ inside $\\mathbb{P}(5 \\mathcal{O} \\oplus \\mathcal{O})$.\n\n\\hypertarget{Fano4--3}{\\subsection*{Fano 4--3}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1$ in a curve of degree $(1,1,2)$.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,1,0,1) \\oplus \\mathcal{O} (0,0,1,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^2.$\n\n\\subsubsection*{Identification} A complete intersection of divisors of degree $(1,1,0)$, $(1,1,1)$ is a curve of degree $(1,1,2)$ in $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1$. In order to blow it up, we use Lemma \\ref{lem:blowDegeneracyLocus}: our Fano is then the zero locus of $\\mathcal{O}(1) \\otimes \\pi^*\\mathcal{O}(1,1,0)$ over $\\pi:\\mathbb{P}(\\mathcal{O}(0,0,-1) \\oplus \\mathcal{O})\\rightarrow \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1$. From the standard Euler sequence on $\\mathbb{P}^1$ we get\n\\[\n0 \\rightarrow\n\\mathcal{O}(0,0,-1) \\oplus \\mathcal{O} \\rightarrow\n\\mathcal{O}^{\\oplus 3} \\rightarrow\n\\mathcal{O}(0,0,1) \\rightarrow\n0,\n\\]\nhence the conclusion.\n\n\n\\hypertarget{Fano4--4}{\\subsection*{Fano 4--4}}\n\\subsubsection*{Mori-Mukai} Blow up of \\hyperlink{Fano3--19}{3--19} in the proper transform of a conic through the points.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^2}(0,1,0) \\oplus \\mathcal{O}(0,2,0) \\oplus \\mathcal{O}(1,0,1)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^4 \\times \\mathbb{P}^1. $\n\n\\subsubsection*{Identification} The first two bundles on $\\mathbb{P}^2 \\times \\mathbb{P}^4$ give the Fano \\hyperlink{Fano3--19}{3--19}. We then just need to use Lemma \\ref{lem:blowup}, since two sections of $\\mathcal{O}(1,0)$ cut the three dimensional quadric in a conic passing through the two points (see also the argument used for \\hyperlink{Fano3--16}{3--16}).\n\n\\hypertarget{Fano4--5}{\\subsection*{Fano 4--5}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{P}^1 \\times \\mathbb{P}^2$ in the disjoint union of a curve of degree $(2,1)$ and a curve of degree $(1,0)$.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(0,1,1,0) \\oplus \\Lambda(0,0,1,0) \\oplus \\mathcal{O}(0,1,0,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^6 \\times \\mathbb{P}^1 $, where the bundle $\\Lambda \\in \\Ext^1(\\mathcal{Q}_{\\mathbb{P}^2}^{\\oplus 2},\\mathcal{O}(1,-1))$ is a uniquely defined extension on $\\mathbb{P}^1 \\times \\mathbb{P}^2$ fitting into sequence \\eqref{Lambda2-18}.\n\n\\subsubsection*{Identification} The first two bundles describe, on $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^6$, the variety \\hyperlink{Fano3--21}{3--21}. We need to blow it up along a curve of degree $(1,0)$, which is the complete intersection of two divisors of degree $(0,1)$ on $\\mathbb{P}^1 \\times \\mathbb{P}^2$. The result follows from Lemma \\ref{lem:blowup}.\n\n\\hypertarget{Fano4--6}{\\subsection*{Fano 4--6}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{P}^3$ in the disjoint union of 3 lines.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,1,0,0) \\oplus \\mathcal{O}(1,0,1,0) \\oplus \\mathcal{O}(1,0,0,1)) \\subset \\mathbb{P}^3 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1. $\n\n\\subsubsection*{Identification} It suffices to apply three times Lemma \\ref{lem:blowup}. By dimension reasons the three lines on $\\mathbb{P}^3$ which are cut each times are disjoint.\n\n\\hypertarget{Fano4--7}{\\subsection*{Fano 4--7}}\n\\subsubsection*{Mori-Mukai} \t\nBlow up of 2--32 in the disjoint union of a curve of degree $(0,1)$ and a curve of degree $(1,0)$.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,0; 1,0) \\oplus \\mathcal{O}(0,1;0,1)) \\subset \\Fl(1,2,3) \\times \\mathbb{P}^1 \\times \\mathbb{P}^1. $\n\n\\subsubsection*{Identification} The flag variety $F:=\\Fl(1,2,3)$ can be identified with 2--32, that is a $(1,1)$ section of $\\mathbb{P}^2 \\times (\\mathbb{P}^2)^{\\vee}.$ Notice that under this identification the generators of the Picard group of the flag are the restriction of the canonical ones on $\\mathbb{P}^2 \\times (\\mathbb{P}^2)^{\\vee}.$ In particular $H^0(F, \\mathcal{O}_{F}(1,0)) \\cong V_3^{\\vee}$ and $H^0(F, \\mathcal{O}_{F}(0,1)) \\cong V_3$.\nThe zero locus of two sections of $\\mathcal{O}_{F}(1,0)$ is a $(0,1)$ curve, and the opposite holds for $\\mathcal{O}_{F}(0,1)$. We then apply twice Lemma \\ref{lem:blowup} to conclude.\n\nOf course thanks to the above identification and Lemma \\ref{lem:blowup} this Fano can be described as well as \n\\[\n\\mathscr{Z}(\\mathcal{O}(1,1,0,0) \\oplus \\mathcal{O}(1,0,1,0) \\oplus \\mathcal{O}(0,1,0,1)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^2 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1.\n\\]\n\n\\hypertarget{Fano4--8}{\\subsection*{Fano 4--8}}\n\\subsubsection*{Mori-Mukai} Blow up of \\hyperlink{Fano3--31}{3--31} (i.e., $\\mathbb{P}_{\\mathbb{P}^1 \\times \\mathbb{P}^1}(\\mathcal{O}(-1,-1) \\oplus \\mathcal{O})$) in a $(1,1)$-section of the base $\\mathbb{P}^1 \\times \\mathbb{P}^1$, or blow up of $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1$ in a curve of degree $(0,1,1)$.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_2 \\oplus \\mathcal{O}(0,2;0) \\oplus \\mathcal{O}(1,0;1) ) \\subset \\Fl(1,2,5) \\times \\mathbb{P}^1.$\n\n\\subsubsection*{Identification} We use the first description by Mori--Mukai, together with our description of \\hyperlink{Fano3--31}{3--31}. Given this, it suffices to apply Lemma \\ref{lem:blowup}, since the zero locus of two extra copies of $\\mathcal{O}_{F}(1,0)$ on $\\mathscr{Z}(\\mathcal{Q}_2 \\oplus \\mathcal{O}_{F}(0,2)) \\subset F:=\\Fl(1,2,5)$ is such a curve. In fact $Z:=\\mathscr{Z}(\\mathcal{Q}_2 \\oplus \\mathcal{O}_{F}(0,2) \\oplus \\mathcal{O}_{F}(1,0)) \\subset F$ corresponds to the base $\\mathbb{P}^1 \\times \\mathbb{P}^1$; on $Z$, both the restrictions $\\mathcal{O}_{F}(1,0)|_Z \\cong \\mathcal{O}_{F}(0,1)|_Z $ coincide with $\\mathcal{O}_{\\mathbb{P}^1 \\times \\mathbb{P}^1}(1,1)$, as can be easily checked via a Chern classes computation.\n\nAlternatively, we can use Lemma \\ref{lem:blowup} to give another description of this Fano, given the alternative one for \\hyperlink{Fano3--31}{3--31}. In particular \\hyperlink{Fano4--8}{4--8} will be given as\n\\[\n\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^3}(0,1,0) \\oplus \\mathcal{O}(2,0,0) \\oplus \\mathcal{O}(0,1,1)) \\subset \\mathbb{P}^3 \\times \\mathbb{P}^4 \\times \\mathbb{P}^1.\n\\]\n\\hypertarget{Fano4--9}{\\subsection*{Fano 4--9}}\n\\subsubsection*{Mori-Mukai} Blow up of \\hyperlink{Fano3--25}{3--25} in an exceptional rational curve $E$ of the blow up.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{Q}_{\\mathbb{P}^2}(0,1,0,0) \\oplus \\mathcal{O}(1,0,1,0) \\oplus \\mathcal{O}(0,1,0,1)) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^3 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1.$\n\n\\subsubsection*{Identification} First we use that the bundle $\\mathcal{Q}_{\\mathbb{P}^2} (0,1) \\subset \\mathbb{P}^2 \\times \\mathbb{P}^3$ gives the blow up $\\Bl_p \\mathbb{P}^3$ by Lemma \\ref{lem:blow}. Lemma \\ref{lem:blowup} yields that the other two bundles yield the blow up along two other lines $L, L'$ in $\\mathbb{P}^3$: $L$ (corresponding to $\\mathcal{O}(1,0,1,0)$) passing through $p$, $L'$ avoiding it (see, e.g., the argument used for \\hyperlink{Fano3--16}{3--16}). Therefore we identify the above variety with $\\Bl_{\\Sigma} \\mathbb{P}^3$, where $\\Sigma:= L \\cup L' \\cup p$, and $p \\in L$. This is the same as $\\Bl_{E}(\\Bl_{L \\cup L'} \\mathbb{P}^3)$. Since the exceptional divisor of the second blow up $\\pi_2$ is a $\\mathbb{P}^1$-bundle over the union of the two lines, (with $E=\\pi_2^{-1}(p)$) the result follows.\n\n\\hypertarget{Fano4--10}{\\subsection*{Fano 4--10}}\n\\subsubsection*{Mori-Mukai} $\\mathbb{P}^1 \\times \\Bl_2 \\mathbb{P}^2$.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(0,1,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(0,0,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^3$.\n\n\n\n\\subsubsection*{Identification} Lemma \\ref{lem:blow} identifies the zero locus of a general section of the second bundle with $\\mathbb{P}^1 \\times \\Bl_p \\mathbb{P}^3$. A section of the remaining bundle gives a quadric in $\\mathbb{P}^3$ containing $p$ (see, e.g., the argument used for \\hyperlink{Fano2--19}{2--19}), which identifies our model with $\\Bl_p (\\mathbb{P}^1 \\times \\mathbb{P}^1)$. We remark that Lemma \\ref{lem:blowup} provides another simple model, i.e., the zero locus of $\\mathcal{O}(1,0,1,0)\\oplus \\mathcal{O}(0,1,1,0)$ over $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^1$.\n\n\\hypertarget{Fano4--11}{\\subsection*{Fano 4--11}}\n\\subsubsection*{Mori-Mukai} Blow up of \\hyperlink{Fano3--28}{3--28} in $\\lbrace x\\rbrace \\times E$, $x \\in \\mathbb{P}^1$ and $E$ the $(-1)$-curve.\n\\subsubsection*{Our description}. $\\mathscr{Z}(\\mathcal{O}(0,1,1,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(0,0,0,1) \\oplus \\Lambda(0,0,0,1) \\oplus \\mathcal{O}(0,0,-1,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^1 \\times \\mathbb{P}^6$, being $\\Lambda \\in \\Ext^1(\\mathcal{O}(0,1)^{\\oplus 2}, \\mathcal{O}(1, -1))$ a uniquely defined extension on $\\mathbb{P}^1 \\times \\mathbb{P}^1$ fitting into \\eqref{Lambda3-2}.\n\\subsubsection*{Identification}\nBy Lemma \\ref{lem:blow} the first bundle defines, on $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^1$, the Fano \\hyperlink{Fano3--28}{3--28}. By \\cite[Corollary 9.12]{EisenbudHarris3264}, we need to blow up the intersection of a $(1,0,0)$ and a $(0,1,-1)$ divisors. Using Lemma \\ref{lem:blowDegeneracyLocus}, our Fano will be the zero locus of $\\mathcal{O}(1) \\otimes \\pi^*\\mathcal{O}(0,0,-1)$ over the projective bundle $\\pi:\\mathbb{P}(\\mathcal{O}(-1,0,-1) \\otimes \\mathcal{O}(0,-1,0)) \\rightarrow \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^1$.\n\nFor $\\mathcal{O}(-1,0,-1)$ we can pull back sequence \\eqref{inclusion3-2} and get\n\\begin{equation}\n0 \\rightarrow \\mathcal{O}(-1,0,-1) \\rightarrow \\mathcal{O}^{\\oplus 4} \\rightarrow \\Lambda \\rightarrow 0,\n\\end{equation}\nwhere $\\Lambda$ fits into \\eqref{Lambda3-2}. Adding it with the standard Euler sequence on $\\mathbb{P}^2$, we get\n\\[\n0 \\rightarrow \\mathcal{O}(-1,0,-1) \\oplus \\mathcal{O}(0,-1,0) \\rightarrow \\mathcal{O}^{\\oplus 7} \\rightarrow \\Lambda \\oplus \\mathcal{Q}_{\\mathbb{P}^2} \\rightarrow 0,\n\\]\nwhich gives the conclusion.\n\nWe remark that the last bundle in the description should be taken with a caveat, as it has no global sections on the ambient space, but acquires some when restricted to the zero locus of the previous bundles. See Caveat \\ref{caveatBundle}.\n\n\\hypertarget{Fano4--12}{\\subsection*{Fano 4--12}}\n\\subsubsection*{Mori-Mukai} Blow up of \\hyperlink{Fano2--33}{2--33} in the disjoint union of two exceptional lines of the blow up.\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,1,0) \\oplus \\Lambda(0,0,1) \\oplus \\mathcal{O}(-1,1,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^3 \\times \\mathbb{P}^{8}$, being $\\Lambda \\in \\Ext^1(\\mathcal{Q}_{\\mathbb{P}^3}^{\\oplus 2},\\mathcal{O}(1,-1))$ a uniquely defined extension on $\\mathbb{P}^1 \\times \\mathbb{P}^3$ fitting into sequence \\eqref{Lambda4-12} below.\n\\subsubsection*{Identification} By Lemma \\ref{lem:blow} (or Lemma \\ref{lem:blowup}) the first bundle gives, on the first two factors, the blow up $Y$ of $\\mathbb{P}^3$ along a line. We then need to blow up two disjoint lines in the exceptional divisor. By \\cite[Corollary 9.12]{EisenbudHarris3264}, the exceptional divisor is a $(-1,1)$ divisor in $Y$; in order to cut out two lines on it, we have to intersect it with the strict transform of a general quadric hypersurface in $\\mathbb{P}^3$, which is a $(0,2)$ divisor cutting the blown up line in two points.\n\nSummarising, we need to blow $Y$ up along the intersection of the two aforementioned divisors. By Lemma \\ref{lem:blowDegeneracyLocus}, this yields that our Fano variety is the zero locus of $\\pi^* \\mathcal{O}(-1,1) \\otimes \\mathcal{O}(1)$ over the projective bundle $\\pi:\\mathbb{P}(\\mathcal{O}(-1,-1) \\oplus \\mathcal{O}) \\rightarrow \\mathbb{P}^1 \\times \\mathbb{P}^3$.\n\nTo describe this projective bundle, we can argue as in Lemma \\ref{projBundle1-12} or Lemma \\ref{projBundle2-2}: we combine the (pull back of the) two (possibly twisted) Euler sequences\n\\begin{gather*}\n0 \\rightarrow \\mathcal{O}(-1,-1) \\rightarrow \\mathcal{O}(0,-1)^{\\oplus 2} \\rightarrow \\mathcal{O}(1,-1) \\rightarrow 0,\\\\\n0 \\rightarrow \\mathcal{O}(0,-1)^{\\oplus 2} \\rightarrow \\mathcal{O}^{\\oplus 8} \\rightarrow \\mathcal{Q}_{\\mathbb{P}^3}^{\\oplus 2} \\rightarrow 0.\n\\end{gather*}\nWe get\n\\begin{gather}\n\\label{inclusion4-12}\n0 \\rightarrow \\mathcal{O}(-1,-1) \\rightarrow \\mathcal{O}^{\\oplus 8} \\rightarrow \\Lambda \\rightarrow 0,\n\\\\\n\\label{Lambda4-12}\n0 \\rightarrow \\mathcal{O}(1,-1) \\rightarrow \\Lambda \\rightarrow \\mathcal{Q}_{\\mathbb{P}^3}^{\\oplus 2} \\rightarrow 0,\n\\end{gather}\nwhere the rank $7$ bundle $\\Lambda$ is homogeneous, not completely reducible and globally generated, and its space of global sections coincides with $H^0(\\mathbb{P}^3, \n\\mathcal{Q}^{\\oplus 2}) \\cong (V_4)^{\\oplus 2}$. Adding $\\mathcal{O} \\rightarrow \\mathcal{O}$ to \\eqref{inclusion4-12} we get that $\\mathbb{P}(\\mathcal{O}(-1,-1) \\oplus \\mathcal{O})$ is the zero locus of $\\Lambda(0,0,1)$ in $\\mathbb{P}^1 \\times \\mathbb{P}^3 \\times \\mathbb{P}^8$, whence the conclusion.\n\nWe remark that the last bundle in the description should be taken with a caveat, as it has no global sections on the ambient space, but acquires some when restricted to the zero locus of the previous bundles. See Caveat \\ref{caveatBundle}.\n\n\n\\hypertarget{Fano4--13}{\\subsection*{Fano 4--13}}\n\\subsubsection*{Mori-Mukai} Blow up of $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1$ in a curve of degree $(1,3,1)$.\n\\subsubsection*{Our description}\n$\\mathscr{Z}(\\Lambda(0,0,0,1) \\oplus \\mathcal{O}(1,0,1,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^4$, being $\\Lambda \\in \\Ext^1(\\mathcal{O}(0,1)^{\\oplus 2}, \\mathcal{O}(1, -1))$ a uniquely defined extension on $\\mathbb{P}^1 \\times \\mathbb{P}^1$ (the first two copies) fitting into \\eqref{Lambda3-2}.\n\\subsubsection*{Identification}\nThe complete intersection between a $(2,1,1)$ and a $(1,0,1)$ divisors is a curve of degree $(1,3,1)$ in $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1$. In order to blow it up, we use Lemma \\ref{lem:blowDegeneracyLocus}: our Fano $Y$ will be the zero locus of $\\mathcal{O}(1) \\otimes \\pi^*\\mathcal{O}(1,0,1)$ over $\\pi:\\mathbb{P}(\\mathcal{O}(-1,-1,0) \\oplus \\mathcal{O})\\rightarrow \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1$. From \\eqref{inclusion3-2} we get that this projective bundle is the zero locus of $\\Lambda(0,0,0,1)$ over $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^4$, where $\\Lambda$ is a bundle on $\\mathbb{P}^1 \\times \\mathbb{P}^1$ (the first two copies) fitting into \\eqref{Lambda3-2}. The conclusion follows.\n\nAnalogously, we could have used the complete intersection of a $(3,1,0)$ and a $(1,1,0)$ divisors, which is again a curve of degree $(1,3,1)$. A similar argument requires the projective bundle $\\mathbb{P}(\\mathcal{O}(-2,-1,0) \\oplus \\mathcal{O}(0,0,-1))$ and produces a model $Y'$ in $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^8$. If we consider the normal sequence for $Y=\\mathscr{Z}(\\mathcal{F}) \\subset \\mathbb{P}:=\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^4$,\na few cohomology computations via the Koszul complex as described in Section \\ref{computeinvariants} provide that $h^0(T_{\\mathbb{P}}|_Y)=33, h^0(\\mathcal{F}|_Y)=34$ and the higher cohomology groups vanish. In \\cite[Lemma 8.11]{pcs} it is shown that the family of curves of degree $(1,1,3)$ on $(\\mathbb{P}^1)^3$ has dimension one (up to the action of $\\Aut((\\mathbb{P}^1)^3)$), and that for all but one curve the automorphism group is finite. This means that a general model $Y$ admits a $(34-33=1)$-dimensional family of deformations, which is the dimension of the moduli of Fano \\hyperlink{Fano4--13}{4--13}, hence $Y$ is general in moduli. The corresponding computations for $Y'\\subset \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^8$ give, analogously, $73-72=1$, so that the models $Y'$ are also general in the moduli space of Fano \\hyperlink{Fano4--13}{4--13}.\n\n\n\\hypertarget{Fano5--1}{\\subsection*{Fano 5--1}}\n\\subsubsection*{Mori-Mukai} Blow up of \\hyperlink{Fano2--29}{2--29} in the disjoint union of three exceptional lines of the blow up.\n\\subsubsection*{Our description}\n$\\mathscr{Z}(\\mathcal{O}(1,1,0,0) \\oplus \\Lambda(0,0,1,0) \\oplus \\mathcal{O}(-1,1,1,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^3}(0,0,0,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^8}(0,0,0,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^3 \\times \\mathbb{P}^{8} \\times \\mathbb{P}^{11}$, being $\\Lambda \\in \\Ext^1(\\mathcal{Q}_{\\mathbb{P}^3}^{\\oplus 2},\\mathcal{O}(1,-1))$ a uniquely defined extension on $\\mathbb{P}^1 \\times \\mathbb{P}^3$ fitting into sequence \\eqref{Lambda4-12}.\n\n\\subsubsection*{Identification} This Fano variety is the blow up of \\hyperlink{Fano4--12}{4--12} along a rational curve. If we consider the model for \\hyperlink{Fano4--12}{4--12} in $\\mathbb{P}^1 \\times \\mathbb{P}^3 \\times \\mathbb{P}^8$ (given by the zero locus of the first three bundles), we can check that the intersection of a $(0,1,0)$ divisor and a $(0,0,1)$ divisor is indeed a rational curve $C$, and the corresponding blow up $Y$ can be checked to have the right Hodge diamond and invariants. To ensure that $Y$ is Fano (hence, it is \\hyperlink{Fano5--1}{5--1}) we can check that a fiber $F$ of the exceptional divisor has $F.K_Y=-1$, so that $-K_Y$ is ample by \\cite[Thm 1.4.3]{isp5}.\n\nAs usual, we blow up $C$ via Lemma \\ref{lem:blowDegeneracyLocus}: our Fano will be the zero locus of $\\mathcal{O}(1)$ over $\\mathbb{P}(\\mathcal{O}(0,-1,0) \\oplus \\mathcal{O}(0,0,-1))$. This projective bundle can be easily described by considering the direct sum of the two Euler sequences, which yields\n\\[\n0 \\rightarrow \\mathcal{O}(0,-1,0) \\oplus \\mathcal{O}(0,0,-1) \\rightarrow \\mathcal{O}^{\\oplus 13} \\rightarrow \\mathcal{Q}_{\\mathbb{P}^3} \\oplus \\mathcal{Q}_{\\mathbb{P}^8} \\rightarrow 0.\n\\]\nThe conclusion follows.\n\n\\hypertarget{Fano5--2}{\\subsection*{Fano 5--2}}\n\\subsubsection*{Mori-Mukai} Blow up of \\hyperlink{Fano3--25}{3--25} in the disjoint union of two exceptional lines on the same irreducible component.\n\n\n\\subsubsection*{Our description} $\\mathscr{Z}(\\mathcal{O}(1,1,0,0) \\oplus \\Lambda(0,0,1,0) \\oplus \\mathcal{O}(-1,1,1,0) \\oplus \\mathcal{O}(0,1,0,1)) \\subset \\mathbb{P}^1 \\times \\mathbb{P}^3 \\times \\mathbb{P}^{8} \\times \\mathbb{P}^1$, being $\\Lambda \\in \\Ext^1(\\mathcal{Q}_{\\mathbb{P}^3}^{\\oplus 2},\\mathcal{O}(1,-1))$ a uniquely defined extension on $\\mathbb{P}^1 \\times \\mathbb{P}^3$ fitting into sequence \\eqref{Lambda4-12}.\n\n\\subsubsection*{Identification} \nRecall that \\hyperlink{Fano4--12}{4--12} is the blow up of $\\mathbb{P}^3$ in a line and then in the disjoint union of two exceptional lines of the blow up, and is given by the zero locus of the first three bundles. To get \\hyperlink{Fano5--2}{5--2} we need to blow it up along the strict transform of a line not intersecting any of the other three. The previously found model for \\hyperlink{Fano4--12}{4--12} was in $\\mathbb{P}^1 \\times \\mathbb{P}^3 \\times \\mathbb{P}^{8}$, and such a line is the complete intersection of two $(0,1,0)$ divisors. Lemma \\ref{lem:blowup} yields the conclusion.\n\n\n\\section{Tables}\n\\label{tables}\nIn this last section we collect in an exhaustive table all the models for Fano 3-folds we exhibited in Section \\ref{Fano3folds}, together with the models already existing in the literature. In Table \\ref{tab:3folds}, MM stands for the Mori--Mukai numeration; the Picard rank $\\rho$ is the first number. In the column ``Inv'' an entry $(a,b,c)$ means the invariants $(h^0(-K), K^3, h^{2,1})$ of the corresponding Fano. The column $X$ refers to the ambient variety, whereas $\\mathcal{F}$ is the bundle whose zero locus produces the 3-fold. In some cases alternative descriptions (marked by ``\\emph{alt.}'') are given, whenever we find them equally interesting. In the column ``Notes'' we put either the reference for the chosen model, when it was not provided by us, or a further explanation of the bundles appearing in the previous column.\n\nWe include a second table, Table \\ref{tab:delpezzo}, for Del Pezzo surfaces, whose models can be easily figured out from Table \\ref{tab:3folds}. Each family in the table (except 2--1) will correspond to the blow up of $\\mathbb{P}^2$ in $9-K^2$ points in sufficiently general position. All models (for 3-folds and Del Pezzo surfaces) are general.\n\n\\begin{centering}\n\\begin{scriptsize}\n\\setlength\\tabcolsep{4pt}\n\\begin{longtable}{ccccc}\n\n\\caption{Fano 3-folds.}\\label{tab:3folds}\\\\\n\\toprule\nMM&\nInv& \n$X$ & \n$\\mathcal{F}$ & \nNotes \\\\\n\\cmidrule(lr){1-1}\\cmidrule(lr){2-2}\\cmidrule(lr){3-3} \\cmidrule(lr){4-4} \\cmidrule(lr){5-5}\n\\endfirsthead\n\\multicolumn{5}{l}{\\vspace{-0.25em}\\scriptsize\\emph{\\tablename\\ \\thetable{} continued from previous page}}\\\\\n\\toprule\nMM&\nInv& \n$X$ & \n$\\mathcal{F}$ & \nNotes \\\\\n\\cmidrule(lr){1-1}\\cmidrule(lr){2-2}\\cmidrule(lr){3-3} \\cmidrule(lr){4-4} \\cmidrule(lr){5-5}\n\\endhead\n\\multicolumn{5}{r}{\\scriptsize\\emph{Continued on next page}}\\\\\n\\endfoot\n\\bottomrule\n\\endlastfoot\n\n\\hyperlink{Fano1--1}{1--1} & $(4,2,52)$ &\\ $\\mathbb{P}(1^4,3)$\\ & $\\mathcal{O}(6)$&\\cite{isp5}\\\\\n\\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}}& $\\mathbb{P}^3 \\times \\mathbb{P}^{20}$&$\\mathcal{O}(0,2) \\oplus K(0,1)$&$K \\in \\Ext^2_{\\mathbb{P}^3}(\\Sym^3 \\mathcal{Q}, \\mathcal{Q}(-2))$\\\\\n\\rowcolor[gray]{0.95} 1--2 & $(5,4,30)$& $\\mathbb{P}^4$& $\\mathcal{O}(4)$& \\cite{isp5} \\\\\n1--3 & $(6,6,20)$& $\\mathbb{P}^5$& $\\mathcal{O}(2) \\oplus \\mathcal{O}(3)$& \\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} 1--4 & $(7,8,14)$& $\\mathbb{P}^6$& $\\mathcal{O}(2)^{\\oplus 3}$&\\cite{isp5} \\\\\n1--5 & $(8,10,10)$& $\\Gr(2,5)$& $\\mathcal{O}(2) \\oplus \\mathcal{O}(1)$&\\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} 1--6 & $(9,12,7)$& $\\OGr^+(5,10)$& $\\mathcal{O}(\\frac{1}{2})^{7}$&\\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\Gr(2,5)$& $\\mathcal{U}^{\\vee}(1)\\oplus \\mathcal{O}(1)$&\\cite{corti} \\\\\n1--7 & $(10,14,5)$& $\\Gr(2,6)$&\n$\\mathcal{O}(1)^5$&\\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} 1--8 & $(11,16,3)$& $\\Gr(3,6)$& $\\bigwedge^2\\mathcal{U}^{\\vee} \\oplus \\mathcal{O}(1)^{3}$&\\cite{isp5} \\\\\n1--9 & $(12,18,2)$& $\\Gr(2,7)$& $\\mathcal{Q}^{\\vee}(1) \\oplus \\mathcal{O}(1)$&\\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} 1--10 & $(14,22,0)$& $\\Gr(3,7)$& $(\\bigwedge^2\\mathcal{U}^{\\vee})^{\\oplus 3}$&\\cite{isp5} \\\\\n1--11 & $(7,8,21)$& $\\mathbb{P}(1^3,2,3)$& $\\mathcal{O}(6)$&\\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} \\hyperlink{Fano1--12}{1--12} & $(11,16,10)$& $\\mathbb{P}(1^4,2)$& $\\mathcal{O}(4)$&\\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\mathbb{P}^3 \\times \\mathbb{P}^{10}$& $\\Lambda(0,1) \\oplus \\mathcal{O}(0,2)$&$\\Lambda \\in \\Ext^1_{\\mathbb{P}^3}(\\Sym^2 \\mathcal{Q}, \\mathcal{Q}(-1))$ \\\\\n1--13 & $(15,24,5)$& $\\mathbb{P}^4$& $\\mathcal{O}(3)$&\\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} 1--14 & $(19,32,2)$& $\\mathbb{P}^5$& $\\mathcal{O}(2)^{\\oplus 2}$&\\cite{isp5} \\\\\n1--15 & $(23,40,0)$& $\\Gr(2,5)$& $\\mathcal{O}(1)^{\\oplus 3}$&\\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} 1--16 & $(30,54,0)$& $\\mathbb{P}^4$& $\\mathcal{O}(2)$&\\cite{isp5} \\\\\n1--17 & $(35,64,0)$& $\\mathbb{P}^3$& & \\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} 2--1 & $(5,4,22)$& $\\mathbb{P}(1^3,2,3) \\times \\mathbb{P}^1$& $\\mathcal{O}(6,0) \\oplus \\mathcal{O}(1,1)$&\\cite{corti} \\\\\n\\hyperlink{Fano2--2}{2--2} & $(6,6,20)$& $\\mathbb{P}^1\\times \\mathbb{P}^2 \\times \\mathbb{P}^{12}$& $\\mathcal{O}(0,0,2) \\oplus K(0,0,1)$&$K \\in \\Ext_{\\mathbb{P}^1 \\times \\mathbb{P}^2}^2(\\mathcal{O}(1,0)^{\\oplus 6}, \\mathcal{Q}_{\\mathbb{P}^2}(-1,-1))$ \\\\\n\\rowcolor[gray]{0.95} \\hyperlink{Fano2--3}{2--3} & $(7,8,11)$& $\\mathbb{P}(1^4,2) \\times \\mathbb{P}^1$& $\\mathcal{O}(4,0) \\oplus \\mathcal{O}(1,1)$&\\cite{corti} \\\\\n\\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\mathbb{P}^3 \\times \\mathbb{P}^{10} \\times \\mathbb{P}^1$& $\\Lambda(0,1,0) \\oplus \\mathcal{O}(0,2,0) \\oplus \\mathcal{O}(1,0,1)$&$\\Lambda \\in \\Ext^1_{\\mathbb{P}^3}(\\Sym^2 \\mathcal{Q}, \\mathcal{Q}(-1))$ \\\\\n2--4 & $(8,10,10)$& $\\mathbb{P}^1 \\times \\mathbb{P}^3$& $\\mathcal{O}(1,3)$&\\cite{corti} \\\\\n\\rowcolor[gray]{0.95} \\hyperlink{Fano2--5}{2--5} & $(9,12,6)$& $\\mathbb{P}^1 \\times \\mathbb{P}^4$& $\\mathcal{O}(0,3) \\oplus \\mathcal{O}(1,1)$& \\\\\n2--6 & $(9,12,9)$& $\\mathbb{P}^2 \\times \\mathbb{P}^2$& $\\mathcal{O}(2,2)$&\\cite{corti} \\\\\n\\rowcolor[gray]{0.95} 2--7 & $(10,14,5)$& $\\mathbb{P}^1 \\times \\mathbb{P}^4$& $\\mathcal{O}(0,2) \\oplus \\mathcal{O}(1,2)$&\\cite{corti} \\\\\n\\hyperlink{Fano2--8}{2--8} & $(10,14,9)$& $\\mathbb{P}^2 \\times \\mathbb{P}^3 \\times \\mathbb{P}^{12}$& $\\Lambda(0,0,1) \\oplus \\mathcal{O}(0,0,2)$&$\\Lambda \\in \\Ext^1_{\\mathbb{P}^2 \\times \\mathbb{P}^3}(\\mathcal{Q}_{\\mathbb{P}^3}^{\\oplus 3}, \\mathcal{Q}_{\\mathbb{P}^2}(0, -1))$ \\\\\n\\rowcolor[gray]{0.95} 2--9 & $(11,16,5)$& $\\mathbb{P}^2 \\times \\mathbb{P}^3$& $\\mathcal{O}(1,1) \\oplus \\mathcal{O}(1,2)$& \\cite{corti} \\\\\n\\hyperlink{Fano2--10}{2--10} & $(11,16,3)$& $\\Gr(2,4) \\times \\mathbb{P}^1$& $\\mathcal{O}(2,0) \\oplus \\mathcal{O}(1,1)$& \\\\\n\\rowcolor[gray]{0.95} \\hyperlink{Fano2--11}{2--11} & $(12,18,5)$& $\\mathbb{P}^2 \\times \\mathbb{P}^4$& $\\mathcal{Q}_{\\mathbb{P}^2}(0,1) \\oplus \\mathcal{O}(1,2)$& \\\\\n\\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\Fl(1,3,5)$& $\\mathcal{Q}_2^{\\oplus 2} \\oplus \\mathcal{O}(2,1)$& \\\\\n2--12 & $(13,20,3)$& $\\mathbb{P}^3 \\times \\mathbb{P}^3$& $\\mathcal{O}(1,1)^{\\oplus 3}$&\\cite{corti} \\\\\n\\rowcolor[gray]{0.95} 2--13 & $(13,20,2)$& $\\mathbb{P}^2 \\times \\mathbb{P}^4$& $\\mathcal{O}(1,1)^{\\oplus 2} \\oplus \\mathcal{O}(0,2)$&\\cite{corti} \\\\\n2--14 & $(13,20,1)$& $\\Gr(2,5) \\times \\mathbb{P}^1$& $\\mathcal{O}(1,0)^{\\oplus 3} \\oplus \\mathcal{O}(1,1)$&\\cite{corti} \\\\\n\\rowcolor[gray]{0.95} \\hyperlink{Fano2--15}{2--15} & $(14,22,4)$& $\\mathbb{P}^3 \\times \\mathbb{P}^4$& $\\mathcal{Q}_{\\mathbb{P}^3}(0,1) \\oplus \\mathcal{O}(2,1)$& \\\\\n\\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\Fl(1,2,5)$& $\\mathcal{Q}_2 \\oplus \\mathcal{O}(1,2)$& \\\\\n \\hyperlink{Fano2--16}{2--16} & $(14,22,2)$& $\\mathbb{P}^2 \\times \\Gr(2,4)$& $\\mathcal{U}^{\\vee}_{\\Gr(2,4)}(1,0) \\oplus \\mathcal{O}(0,2)$& \\\\\n \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\Fl(1,2,4)$& $\\mathcal{O}(1,0) \\oplus \\mathcal{O}(0,2)$& \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano2--17}{2--17} & $(15,24,1)$& $\\Gr(2,4) \\times \\mathbb{P}^3$& $\\mathcal{U}_{\\Gr(2,4)}^{\\vee}(0,1) \\oplus \\mathcal{O}(1,1) \\oplus \\mathcal{O}(1,0)$&\\cite{corti} \\\\\n \\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\Fl(1,2,4)$& $\\mathcal{O}(0,1) \\oplus \\mathcal{O}(1,1)$& \\\\\n \\hyperlink{Fano2--18}{2--18} & $(15,24,2)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^6$& $\\Lambda(0,0,1) \\oplus \\mathcal{O}(0,0,2)$&$\\Lambda \\in \\Ext^1_{\\mathbb{P}^1 \\times \\mathbb{P}^2}(\\mathcal{Q}_{\\mathbb{P}^2}^{\\oplus 2}, \\mathcal{O}(1,-1))$ \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano2--19}{2--19} & $(16,26,2)$& $\\mathbb{P}^3 \\times \\mathbb{P}^5$& $\\mathcal{Q}_{\\mathbb{P}^3}(0,1) \\oplus \\mathcal{O}(1,1)^{\\oplus 2}$& \\\\\n \\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\Fl(1,3,6)$& $\\mathcal{Q}_2^{\\oplus 2} \\oplus \\mathcal{O}(1,1)^{\\oplus 2}$& \\\\\n 2--20 & $(16,26,0)$& $\\Gr(2,5) \\times \\mathbb{P}^2$& $\\mathcal{U}^{\\vee}_{\\Gr(2,5)}(0,1) \\oplus \\mathcal{O}(1,0)^{\\oplus 3}$&\\cite{corti} \\\\\n \\rowcolor[gray]{0.95} 2--21 & $(17,28,0)$& $\\Gr(2,4) \\times \\mathbb{P}^2$& $\\mathcal{U}^{\\vee}_{\\Gr(2,4)}(0,1)^{\\oplus 2} \\oplus \\mathcal{O}(1,0)$&\\cite{corti} \\\\\n \\hyperlink{Fano2--22}{2--22} & $(18,30,0)$& $\\mathbb{P}^3 \\times \\Gr(2,5)$& $\\mathcal{Q}_{\\Gr(2,5)}(1,0) \\oplus \\mathcal{O}(0,1)^{\\oplus 3}$& \\\\\n \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\Fl(1,2,5)$& $\\mathcal{O}(1,0) \\oplus \\mathcal{O}(0,1)^{\\oplus 3}$&\\cite{corti} \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano2--23}{2--23} & $(18,30,1)$& $\\mathbb{P}^4 \\times \\mathbb{P}^5$& $\\mathcal{Q}_{\\mathbb{P}^4}(0,1) \\oplus \\mathcal{O}(2,0) \\oplus \\mathcal{O}(1,1)$& \\\\*\n \\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\Fl(1,2,6)$& $\\mathcal{Q}_2 \\oplus \\mathcal{O}(0,2) \\oplus \\mathcal{O}(1,1)$& \\\\\n 2--24 & $(18,30,0)$& $\\mathbb{P}^2 \\times \\mathbb{P}^2$& $\\mathcal{O}(1,2)$& \\cite{isp5} \\\\ \n \\rowcolor[gray]{0.95} 2--25 & $(19,32,1)$& $\\mathbb{P}^1 \\times \\mathbb{P}^3$& $\\mathcal{O}(1,2)$&\\cite{corti} \\\\\n \\hyperlink{Fano2--26}{2--26} & $(20,34,0)$& $\\Gr(2,4) \\times \\Gr(2,5)$& $\\mathcal{Q}_{\\Gr(2,4)} \\boxtimes \\mathcal{U}_{\\Gr(2,5)}^{\\vee} \\oplus \\mathcal{O}(1,0) \\oplus \\mathcal{O}(0,1)^{\\oplus 2}$& \\\\\n \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\Fl(2,3,5)$& $\\mathcal{U}_1^{\\vee} \\oplus \\mathcal{O}(1,0) \\oplus \\mathcal{O}(0,1)^{\\oplus 2}$& \\\\\n \\rowcolor[gray]{0.95} 2--27 & $(22,38,0)$& $\\mathbb{P}^3 \\times \\mathbb{P}^2$& $\\mathcal{O}(1,1)^{\\oplus 2}$&\\cite{corti} \\\\\n \\hyperlink{Fano2--28}{2--28} & $(23,40,1)$& $\\mathbb{P}^3 \\times \\mathbb{P}^{10}$& $\\Lambda(0,1) \\oplus \\mathcal{O}(1,1)$&$\\Lambda \\in \\Ext^1_{\\mathbb{P}^3}(\\Sym^2\\mathcal{Q}, \\mathcal{Q}(-1))$ \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano2--29}{2--29} & $(23,40,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^4$& $\\mathcal{O}(0,2) \\oplus \\mathcal{O}(1,1)$& \\\\\n \\hyperlink{Fano2--30}{2--30} & $(26,46,0)$& $\\mathbb{P}^3 \\times \\mathbb{P}^4$& $\\mathcal{Q}_{\\mathbb{P}^3}(0,1) \\oplus \\mathcal{O}(1,1)$& \\\\\n \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\Fl(1,2,5)$& $\\mathcal{Q}_2 \\oplus \\mathcal{O}(1,1)$& \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano2--31}{2--31} & $(26,46,0)$& $\\mathbb{P}^2 \\times \\Gr(2,4)$& $\\mathcal{U}^{\\vee}_{\\Gr(2,4)}(1,0) \\oplus \\mathcal{O}(0,1)$& \\\\\n 2--32 & $(27,48,0)$& $\\mathbb{P}^2 \\times \\mathbb{P}^2$& $\\mathcal{O}(1,1)$&\\cite{isp5} \\\\\n \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}}& $\\Fl(1,2,3)$& &\\cite{isp5} \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano2--33}{2--33} & $(30,54,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^3$& $\\mathcal{O}(1,1)$& \\\\\n 2--34 & $(30,54,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2$& &\\cite{isp5} \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano2--35}{2--35} & $(31,56,0)$& $\\mathbb{P}^2 \\times \\mathbb{P}^3$& $\\mathcal{Q}_{\\mathbb{P}^2}(0,1)$& \\\\\n \\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\Fl(1,2,4)$& $\\mathcal{Q}_2$& \\\\\n \\hyperlink{Fano2--36}{2--36} & $(34,62,0)$& $\\mathbb{P}^2 \\times \\mathbb{P}^6$& $\\Lambda(0,1)$&$\\Lambda \\in \\Ext^1_{\\mathbb{P}^2}(\\Sym^2\\mathcal{Q}, \\mathcal{Q}(-1))$ \\\\\n\n \n \\rowcolor[gray]{0.95} \\hyperlink{Fano3--1}{3--1} & $(9,12,8)$& $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^8$& $K(0,0,0,1) \\oplus \\mathcal{O}(0,0,0,2)$&$K \\in \\Ext^2_{(\\mathbb{P}^1)^3}(\\mathcal{O}(0,0,1)^{\\oplus 4},\\mathcal{O}(1,-1,-1))$ \\\\\n \\hyperlink{Fano3--2}{3--2} & $(10,14,3)$& $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^5$& $\\Lambda(0,0,1) \\oplus \\mathcal{O}(0,1,2)$&$\\Lambda \\in \\Ext^1_{(\\mathbb{P}^1)^2}(\\mathcal{O}(0,1)^{\\oplus 2}, \\mathcal{O}(1, -1))$ \\\\\n \\rowcolor[gray]{0.95} 3--3 & $(12,18,3)$& $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^2$& $\\mathcal{O}(1,1,2)$&\\cite{corti} \\\\\n \\hyperlink{Fano3--4}{3--4} & $(12,18,2)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^6 \\times \\mathbb{P}^1$& $\\Lambda(0,0,1,0) \\oplus \\mathcal{O}(0,0,2,0) \\oplus \\mathcal{O}(0,1,0,1)$&$\\Lambda \\in \\Ext^1_{\\mathbb{P}^1 \\times \\mathbb{P}^2}(\\mathcal{Q}_{\\mathbb{P}^2}^{\\oplus 2}, \\mathcal{O}(1,-1))$ \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano3--5}{3--5} & $(13,20,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^7$& $\\Lambda(0,0,1) \\oplus \\mathcal{O}(0,1,1)^{\\oplus 2}$&$\\Lambda \\in \\Ext^1_{\\mathbb{P}^1 \\times \\mathbb{P}^2}(\\mathcal{Q}_{\\mathbb{P}^2}^{\\oplus 2}, \\mathcal{O}(1,-1))$ \\\\\n \\hyperlink{Fano3--6}{3--6} & $(14,22,1)$& $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^3$& $\\mathcal{O}(1,0,2) \\oplus \\mathcal{O}(0,1,1)$& \\\\\n \\rowcolor[gray]{0.95} 3--7 & $(15,24,1)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^2$& $\\mathcal{O}(0,1,1) \\oplus \\mathcal{O}(1,1,1)$&\\cite{corti} \\\\\n \\hyperlink{Fano3--8}{3--8} & $(15,24,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^2$& $\\mathcal{O}(0,1,2) \\oplus \\mathcal{O}(1,1,0)$& \\\\\n \\rowcolor[gray]{0.95} & & & &\\multicolumn{1}{c}{$\\Lambda \\in \\Ext^1_{\\mathbb{P}^2}(\\Sym^2\\mathcal{Q}, \\mathcal{Q}(-1)),$} \\\\\n \\rowcolor[gray]{0.95} \\multirow{-2}{*}{\\hyperlink{Fano3--9}{3--9}}& \\multirow{-2}{*}{$(16,26,3)$} & \\multirow{-2}{*}{$\\mathbb{P}^2 \\times \\mathbb{P}^6 \\times \\mathbb{P}^{20}$} & \\multirow{-2}{*}{$\\Lambda(0,1,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^6}(0,0,1) \\oplus K(0,0,1)$}& \\multicolumn{1}{c}{$K \\in \\Ext^3_{\\mathbb{P}^2}(\\Sym^4\\mathcal{Q}, \\mathcal{Q}(-3))$}\\\\\n \\hyperlink{Fano3--10}{3--10} & $(16,26,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^4$& $\\mathcal{O}(1,0,1) \\oplus \\mathcal{O}(0,1,1) \\oplus \\mathcal{O}(0,0,2)$& \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano3--11}{3--11} & $(17,28,1)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^3$& $\\mathcal{O}(1,1,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(0,0,1)$& \\\\\n \\hyperlink{Fano3--12}{3--12} & $(17,28,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^3$& $\\mathcal{O}(0,1,1) \\oplus \\mathcal{O}(0,1,1) \\oplus \\mathcal{O}(1,0,1)$& \\\\\n \\rowcolor[gray]{0.95} 3--13 & $(18,30,0)$& $(\\mathbb{P}^2)^3$& $\\mathcal{O}(1,1,0) \\oplus \\mathcal{O}(1,0,1) \\oplus(0,1,1)$&\\cite{corti} \\\\\n \\hyperlink{Fano3--14}{3--14} & $(19,32,1)$& $\\mathbb{P}^3 \\times \\mathbb{P}^{10} \\times \\mathbb{P}^2$& $\\Lambda(0,1,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(1,0,0) \\oplus \\mathcal{O}(1,1,0)$&$\\Lambda \\in \\Ext^1_{\\mathbb{P}^3}(\\Sym^2\\mathcal{Q}, \\mathcal{Q}(-1)) $ \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano3--15}{3--15} & $(19,32,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^4$& $\\mathcal{O}(1,0,1) \\oplus \\mathcal{O}(0,1,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(0,0,1)$& \\\\\n \\hyperlink{Fano3--16}{3--16} & $(20,34,0)$& $\\mathbb{P}^2_1 \\times \\mathbb{P}^2_2 \\times \\mathbb{P}^3$& $\\mathcal{O}(0,1,1) \\oplus \\mathcal{O}(1,1,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^2_1}(0,0,1)$& \\\\\n \\rowcolor[gray]{0.95} 3--17 & $(21,36,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^2$& $\\mathcal{O}(1,1,1)$&\\cite{corti} \\\\\n \\hyperlink{Fano3--18}{3--18} & $(21,36,0)$& $\\mathbb{P}^1\\times \\mathbb{P}^3 \\times \\mathbb{P}^4$& $\\mathcal{O}(1,1,0) \\oplus \\mathcal{O}(0,1,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^3}(0,0,1)$& \\\\\n \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\mathbb{P}^1 \\times \\Fl(1,2,5)$& $\\mathcal{Q}_2(0;0,0) \\oplus \\mathcal{O}(0;1,1) \\oplus \\mathcal{O}(1;0,1)$& \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano3--19}{3--19} & $(22,38,0)$& $\\mathbb{P}^2 \\times \\mathbb{P}^4$& $\\mathcal{O}(0,2) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(0,1)$& \\\\\n \\hyperlink{Fano3--20}{3--20} & $(22,38,0)$& $\\mathbb{P}^2_1 \\times \\mathbb{P}^2_2 \\times \\mathbb{P}^5$& $\\mathcal{O}(1,1,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^2_1}(0,0,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2_2}(0,0,1)$& \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano3--21}{3--21} & $(22,38,0)$& $\n \\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^6$& $\\mathcal{O}(0,1,1) \\oplus \\Lambda(0,0,1)$&$\\Lambda \\in \\Ext^1_{\\mathbb{P}^1 \\times \\mathbb{P}^2}(\\mathcal{Q}_{\\mathbb{P}^2}^{\\oplus 2}, \\mathcal{O}(1,-1)) $ \\\\\n \\hyperlink{Fano3--22}{3--22} & $(23,40,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^6$& $\\mathcal{O}(1,0,1) \\oplus \\Lambda(0,0,1)$&$\\Lambda \\in \\Ext^1_{\\mathbb{P}^2}(\\Sym^2 \\mathcal{Q}, \\mathcal{Q}(-1))$ \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano3--23}{3--23} & $(24,42,0)$& $\\mathbb{P}^2 \\times \\mathbb{P}^3 \\times \\mathbb{P}^4$& $\\mathcal{Q}_{\\mathbb{P}^2}(0,1,0) \\oplus \\mathcal{O}(1,0,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^3}(0,0,1)$& \\\\\n \\hyperlink{Fano3--24}{3--24} & $(24,42,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^2$& $\\mathcal{O}(1,1,0) \\oplus \\mathcal{O}(0,1,1)$&\\cite{corti} \\\\\n\\rowcolor[gray]{0.95} \\hyperlink{Fano3--25}{3--25} & $(25,44,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^3$& $\\mathcal{O}(1,0,1) \\oplus \\mathcal{O}(0,1,1)$& \\\\\n\\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\Fl(1,2,4)$& $\\mathcal{O}(0,1)^{\\oplus 2}$& \\\\\n\\hyperlink{Fano3--26}{3--26} & $(26,46,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^3$& $\\mathcal{O}(1,0,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(0,0,1)$& \\\\\n\\rowcolor[gray]{0.95} 3--27 & $(27,48,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1$& &\\cite{isp5} \\\\\n\\hyperlink{Fano3--28}{3--28} & $(27,48,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^2$& $\\mathcal{O}(1,0,1)$& \\\\\n\\rowcolor[gray]{0.95} & & & \\multicolumn{1}{c}{$\\mathcal{Q}_{\\mathbb{P}^2}(1,0,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^3}(0,0,1) \\oplus$} &\n\\\\\n\\rowcolor[gray]{0.95} \\multirow{-2}{*}{\\hyperlink{Fano3--29}{3--29}} & \\multirow{-2}{*}{$(28,50,0)$}& \\multirow{-2}{*}{$\\mathbb{P}^3 \\times \\mathbb{P}^2 \\times \\mathbb{P}^9$}& \\multicolumn{1}{c}{$\\oplus \\Lambda(0,0,1) \\oplus \\mathcal{O}(0,-1,1)$}&\\multirow{-2}{*}{$\\Lambda \\in \\Ext^1_{\\mathbb{P}^2}(\\Sym^2\\mathcal{Q}, \\mathcal{Q}(-1))$} \\\\\n\\hyperlink{Fano3--30}{3--30} & $(28,50,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^3$& $\\mathcal{O}(1,1,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(0,0,1)$& \\\\\n\\rowcolor[gray]{0.95} \\hyperlink{Fano3--31}{3--31} & $(29,52,0)$& $\\mathbb{P}^3 \\times \\mathbb{P}^4$& $\\mathcal{Q}_{\\mathbb{P}^3}(0,1) \\oplus \\mathcal{O}(2,0)$& \\\\\n\\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\Fl(1,2,5)$& $\\mathcal{Q}_2 \\oplus \\mathcal{O}(0,2)$& \\\\\n4--1 & $(15,24,1)$& $(\\mathbb{P}^1)^4$& $\\mathcal{O}(1,1,1,1)$&\\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} & & & $\\mathcal{Q}_{\\mathbb{P}^3}(0,1,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^4}(0,0,1) \\oplus $ & \\\\\n\\rowcolor[gray]{0.95} \\multirow{-2}{*}{\\hyperlink{Fano4--2}{4--2}} & \\multirow{-2}{*}{$(17,28,1)$}& \\multirow{-2}{*}{$\\mathbb{P}^3 \\times \\mathbb{P}^4 \\times \\mathbb{P}^5$}& $\\oplus \\mathcal{O}(2,0,0) \\oplus \\mathcal{O}(0,1,1) $ &\\\\\n\n \\hyperlink{Fano4--3}{4--3} & $(18,30,0)$& $(\\mathbb{P}^1)^3 \\times \\mathbb{P}^2$& $\\mathcal{O}(1,1,0,1) \\oplus \\mathcal{O}(0,0,1,1)$& \\\\\n \n \\rowcolor[gray]{0.95} \\hyperlink{Fano4--4}{4--4} & $(19,32,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^4$& $\\mathcal{O}(1,1,0) \\oplus \\mathcal{O}(0,0,2) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(0,0,1)$& \\\\\n \\hyperlink{Fano4--5}{4--5} & $(19,32,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^6 \\times \\mathbb{P}^1$& $\\mathcal{O}(0,1,1,0) \\oplus \\Lambda(0,0,1,0) \\oplus \\mathcal{O}(0,1,0,1)$&$\\Lambda \\in \\Ext^1_{\\mathbb{P}^1 \\times \\mathbb{P}^2}(\\mathcal{Q}_{\\mathbb{P}^2}^{\\oplus 2},\\mathcal{O}(1,-1))$ \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano4--6}{4--6} & $(20,34,0)$& $(\\mathbb{P}^1)^3 \\times \\mathbb{P}^3$& $\\mathcal{O}(1,0,0,1) \\oplus \\mathcal{O}(0,1,0,1) \\oplus \\mathcal{O}(0,0,1,1)$& \\\\\n \\hyperlink{Fano4--7}{4--7} & $(21,36,0)$& $(\\mathbb{P}^1)^2 \\times (\\mathbb{P}^2)^2$& $\\mathcal{O}(0,0,1,1) \\oplus \\mathcal{O}(1,0,1,0) \\oplus \\mathcal{O}(0,1,0,1)$&\\\\\n \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $ (\\mathbb{P}^1)^2 \\times \\Fl(1,2,3)$& $\\mathcal{O}(1,0; 1,0) \\oplus \\mathcal{O}(0,1;0,1)$& \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano4--8}{4--8} & $(22,38,0)$& $\\mathbb{P}^1 \\times \\mathbb{P}^3 \\times \\mathbb{P}^4$& $\\mathcal{O}(1,0,1) \\oplus \\mathcal{O}(0,2,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^3}(0,0,1)$& \\\\\n \\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\mathbb{P}^1 \\times \\Fl(1,2,5)$& $\\mathcal{O}(1;1,0) \\oplus \\mathcal{O}(0;0,2) \\oplus \\mathcal{Q}_2$& \\\\\n \\hyperlink{Fano4--9}{4--9} & $(23,40,0)$& $(\\mathbb{P}^1)^2 \\times \\mathbb{P}^2 \\times \\mathbb{P}^3$& $\\mathcal{O}(1,0,1,0) \\oplus \\mathcal{O}(0,1,0,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(0,0,0,1)$& \\\\\n \\rowcolor[gray]{0.95} \\hyperlink{Fano4--10}{4--10} & $(24,42,0)$& $(\\mathbb{P}^1)^3 \\times \\mathbb{P}^2$& $\\mathcal{O}(1,0,0,1) \\oplus \\mathcal{O}(0,1,0,1)$& \\\\\n \\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\mathbb{P}^1\\times \\mathbb{P}^2 \\times \\mathbb{P}^3$& $\\mathcal{O}(0,1,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(0,0,1)$& \\\\\n\n & & & \\multicolumn{1}{c}{$\\mathcal{O}(0,1,1,0) \\oplus \\mathcal{Q}_{\\mathbb{P}^2}(0,0,0,1) \\oplus$}\\\\\n \\multirow{-2}{*}{\\hyperlink{Fano4--11}{4--11}}\n & \\multirow{-2}{*}{$(25,44,0)$}& \\multirow{-2}{*}{$\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^1 \\times \\mathbb{P}^6$}& \\multicolumn{1}{c}{$\\oplus \\Lambda(0,0,0,1) \\oplus \\mathcal{O}(0,0,-1,1)$}&\\multirow{-2}{*}{$\\Lambda \\in \\Ext^1_{(\\mathbb{P}^1)^2}(\\mathcal{O}(0,1)^{\\oplus 2}, \\mathcal{O}(1, -1))$} \\\\\n \n \\rowcolor[gray]{0.95} \\hyperlink{Fano4--12}{4--12} & $(26,46,0)$& $ \\mathbb{P}^1 \\times \\mathbb{P}^3 \\times \\mathbb{P}^{8}$& $\\mathcal{O}(1,1,0) \\oplus \\Lambda(0,0,1) \\oplus \\mathcal{O}(-1,1,1)$&$\\Lambda \\in \\Ext^1_{\\mathbb{P}^1 \\times \\mathbb{P}^3}(\\mathcal{Q}_{\\mathbb{P}^3}^{\\oplus 2},\\mathcal{O}(1,-1))$ \\\\ \n \\hyperlink{Fano4--13}{4--13} & $(16,26,0)$& $ \\mathbb{P}^1_1 \\times \\mathbb{P}^1_2 \\times \\mathbb{P}^1_3 \\times \\mathbb{P}^4$& $\\mathscr{Z}(\\Lambda(0,0,0,1) \\oplus \\mathcal{O}(1,0,1,1)) $&$\\Lambda \\in \\Ext^1_{\\mathbb{P}^1_1 \\times \\mathbb{P}^1_2}(\\mathcal{O}(0,1)^{\\oplus 2}, \\mathcal{O}(1, -1))$ \\\\\n \\rowcolor[gray]{0.95} & & & $\\mathcal{O}(1,1,0,0) \\oplus \\Lambda(0,0,1,0) \\oplus \\mathcal{O}(-1,1,1,0) \\oplus $& \\\\\n \\rowcolor[gray]{0.95} \\multirow{-2}{*}{\\hyperlink{Fano5--1}{5--1}} & \\multirow{-2}{*}{$(17,28,0)$}& \\multirow{-2}{*}{$\\mathbb{P}^1 \\times \\mathbb{P}^3 \\times \\mathbb{P}^{8} \\times \\mathbb{P}^{11}$}& $\\oplus \\mathcal{Q}_{\\mathbb{P}^3}(0,0,0,1) \\oplus \\mathcal{Q}_{\\mathbb{P}^8}(0,0,0,1)$& \\multirow{-2}{*}{$\\Lambda \\in \\Ext^1_{\\mathbb{P}^1 \\times \\mathbb{P}^3}(\\mathcal{Q}_{\\mathbb{P}^3}^{\\oplus 2},\\mathcal{O}(1,-1))$}\\\\\n \n\n& & & $\\mathcal{O}(1,1,0,0) \\oplus \\Lambda(0,0,1,0) \\oplus$\n\\\\\n\\multirow{-2}{*}{\\hyperlink{Fano5--2}{5--2}} & \\multirow{-2}{*}{$(21,36,0)$}& \\multirow{-2}{*}{$\\mathbb{P}^1 \\times \\mathbb{P}^3 \\times \\mathbb{P}^{8} \\times \\mathbb{P}^1$}& $\\oplus \\mathcal{O}(-1,1,1,0) \\oplus \\mathcal{O}(0,1,0,1)$&\\multirow{-2}{*}{$\\Lambda \\in \\Ext^1_{\\mathbb{P}^1 \\times \\mathbb{P}^3}(\\mathcal{Q}_{\\mathbb{P}^3}^{\\oplus 2},\\mathcal{O}(1,-1))$} \\\\\n\\rowcolor[gray]{0.95} 5--3 & $(21,36,0)$& $\\mathbb{P}^2 \\times \\mathbb{P}^2 \\times \\mathbb{P}^1$& $\\mathcal{O}(1,1,0)^{\\oplus 2}$& \\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $(\\mathbb{P}^1)^4$& $\\mathcal{O}(1,1,1,0)$& \\cite{isp5} \\\\\n6--1 & $(18,30,0)$& $\\Gr(2,5) \\times \\mathbb{P}^1$& $\\mathcal{O}(1,0)^{\\oplus 4}$&\\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} 7--1 & $(15,24,0)$& $\\mathbb{P}^4 \\times \\mathbb{P}^1$& $\\mathcal{O}(2,0)^{\\oplus 2}$&\\cite{isp5} \\\\\n8--1 & $(12,18,0)$& $\\mathbb{P}^3 \\times \\mathbb{P}^1$& $\\mathcal{O}(3,0)$&\\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} 9--1 & $(9,12,0)$& $\\mathbb{P}(1^3,2) \\times \\mathbb{P}^1$& $\\mathcal{O}(4,0)$&\\cite{isp5} \\\\\n\\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\mathbb{P}^1 \\times \\mathbb{P}^2 \\times \\mathbb{P}^1$& $\\mathcal{O}(2,2,0)$&\\cite{isp5} \\\\\n10--1 & $(6,10,0)$& $\\mathbb{P}(1^2,2,3) \\times \\mathbb{P}^1$& $\\mathcal{O}(6,0)$&\\cite{isp5} \\\\\n\\end{longtable}\n\\end{scriptsize}\n\\end{centering}\n\n\n\\begin{longtable}{cccccrc}\n\\caption{Del Pezzo surfaces.}\\label{tab:delpezzo}\\\\\n\\toprule\n\\multicolumn{1}{c}{DP}&\\multicolumn{1}{c}{$K^2$}& \\multicolumn{1}{c}{X} & \\multicolumn{1}{c}{$\\mathcal{F}$} \\\\\n\\cmidrule(lr){1-1}\\cmidrule(lr){2-2}\\cmidrule(lr){3-3} \\cmidrule(lr){4-4}\n\\endfirsthead\n\\multicolumn{5}{l}{\\vspace{-0.25em}\\scriptsize\\emph{\\tablename\\ \\thetable{} continued from previous page}}\\\\\n\\midrule\n\\endhead\n\\multicolumn{5}{r}{\\scriptsize\\emph{Continued on next page}}\\\\\n\\endfoot\n\\bottomrule\n\\endlastfoot\n\\rowcolor[gray]{0.95} 1--1 & $9$& $\\mathbb{P}^2$& \\\\\n 2--1 & $8$& $\\mathbb{P}^1 \\times \\mathbb{P}^1$& \\\\\n\\rowcolor[gray]{0.95} 2--2 & $8$& $\\mathbb{P}^1 \\times \\mathbb{P}^2$& $\\mathcal{O}(1,1) $\\\\\n\n3--1 & $7$& $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^2$& $\\mathcal{O}(1,0,1) \\oplus \\mathcal{O}(0,1,1)$\\\\\n\\rowcolor[gray]{0.95} 4--1 & $6$& $\\mathbb{P}^2 \\times \\mathbb{P}^2$& $\\mathcal{O}(1,1)^{\\oplus 2}$\\\\\n\\rowcolor[gray]{0.95} \\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $(\\mathbb{P}^1)^3$& $\\mathcal{O}(1,1,1)$\\\\\n5--1 & $5$& $\\Gr(2,5)$& $\\mathcal{O}(1)^{\\oplus 4}$\\\\\n\\rowcolor[gray]{0.95} 6--1 & $4$& $\\mathbb{P}^4 $& $\\mathcal{O}(2)^{\\oplus 2}$\\\\\n7--1 & $3$& $\\mathbb{P}^3 $& $\\mathcal{O}(3)$\\\\\n\\rowcolor[gray]{0.95} 8--1 & $2$& $\\mathbb{P}(1^3,2)$& $\\mathcal{O}(4)$\\\\\n\\rowcolor[gray]{0.95} \\rowcolor[gray]{0.95} \\multicolumn{2}{l}{\\rule{10pt}{0pt}\\emph{alt.}} & $\\mathbb{P}^1 \\times \\mathbb{P}^2$& $\\mathcal{O}(2,2)$\\\\\n9--1 & $1$& $\\mathbb{P}(1^2,2,3)$& $\\mathcal{O}(6)$\\\\\n\n\\end{longtable}\n\n\\frenchspacing\n\n\n\\newcommand{\\etalchar}[1]{$^{#1}$}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\n\\pagestyle{myheadings}\n\\markboth{\\sc Churchill \\& Vogt \\hfill The Halo: QSO Absorption~~}\n {\\sc Churchill \\& Vogt \\hfill The Halo: QSO Absorption~~}\n\nAs Sidney van den Bergh stated in his closing comments of the meeting,\nthere are two general approaches to studying the history and\nevolution of the Galactic Halo. \nThe first is to study the kinematic, chemical, and structural\ncomponents of the Halo itself, and the second is to study the \nhalos of external galaxies with the partial aim of placing the Halo in\nthe broader context offered by various morphologies and environments\n(cf.~Morrison, this volume).\nGenerally, the tracers of Halo formation (e.g.~globular clusters,\nRR~Lyrae stars, Blue Horizontal Branch stars, star counts, etc.) are\ntreated as frozen relics of past formation processes.\nThe hope is that these processes have left signatures, such as\nkinematic trends or metallicity gradients, which can be used to\nclearly discern between competing formation scenarios.\n\nOne important tracer, which to a certain degree has not captured the\nattention of those studying Milky Way evolution {\\it per se},\nis low density gas in the Halo.\nUsing the absorption lines seen in the spectra of background quasars\n(QSO Absorption Lines, or QALs), several groups have carefully mapped\nout the sky locations of large cloud complexes (cf.~van~Woerden, this\nvolume), including high--velocity clouds (HVCs), which by themselves\nhave a sky covering factor of 38\\%.\nThe connection between HVCs and Halo tracers is not known, though\nMajewski (this volume) has reported stellar moving groups which\ncorrelate well with the locations and velocities of HVCs toward\nthe North Galactic Pole.\nFrom QAL surveys of galaxies at redshifts $0.4 \\leq z \\leq\n2.2$, gaseous halos of normal galaxies are known to extend to $\\sim\n70$~kpc and to contain an estimated $10^{9}$--$10^{10}$~M$_{\\sun}$ of\ngas (Steidel 1993; Steidel \\& Sargent 1992).\nHalo gas is the reservoir for star formation and chemical evolution, \nand plays a central role in the formation of the tracers used to study \nHalo formation. \nIts study in early epoch galaxies promises to yield important\nclues to the processes that regulated galaxy formation\n(cf.~Bechtold, this volume).\n\n\\section*{Learning about the Milky Way from QALs}\n\nQAL studies are unique in that they directly probe galactic gas over\nthe entire history of galaxy evolution.\nThus, they provide a powerful method from which to indirectly\n``view'' Milky Way Halo formation over a multi--billion year period\nand offer a very broad context within which studies using Halo tracers\ncan be placed.\nIn tandem with high spatial resolution imaging (Hubble Space Telescope)\nand high quality spectra of the absorbing galaxies themselves, the\npotential provided by QAL studies can be fully realized.\n\nFrom high resolution QAL spectra (see Fig.~1), we can measure the\nnumber of clouds intercepted in a galaxy, their column densities,\nbroadening mechanisms, and line--of--sight velocities. \nHST images provide the absorbing galaxies' environments, luminosities,\ncolors, impact parameters to the QSO (projected galactocentric\ndistance of the absorbing clouds), and orientations relative to the\nline of sight.\nSpectra of the galaxies can be used to estimate star formation rates,\nand (with LRIS on Keck) obtain rotation curves out to $z \\sim 1$.\nCase by case, we can study the physical details of absorbing gas and\nits relationship to the host galaxy, and then piece together a\ncomprehensive picture of halo evolution directly from the large range of\nepochs the galaxies sample.\nThe unexplored connections between absorption properties and the\nlocations probed in galaxies, their morphologies, redshifts, and\nenvironments will ultimately be used to develop a global picture of\nkinematic and chemical evolution.\n\nAs a first step, we have observed 24 QSOs with the HIRES spectrograph\n(Vogt et~al.~1994). \nWe have resolved absorption profiles of the Mg~{\\sc ii}\n$\\lambda\\lambda2796,2803$ resonant doublet in $\\sim 50$ intervening\ngalaxies.\nMany of these galaxies have been ground--based imaged in the\nIR\/optical and spectroscopically confirmed to have the redshift seen\nin absorption (Steidel 1995; Steidel, Dickinson, \\& Persson 1996).\n\n\n\n\\section*{Gaseous Fragments and Kinematic Evolution}\n\nIn this contribution, we present partial results and a brief discussion\nof work to be published elsewhere (Churchill \\& Vogt 1996).\nIn Fig.~1, we show four HIRES\/Keck Mg~{\\sc ii} absorption profiles\nas seen in the spectra of background QSOs.\nThese absorption lines arise in low ionization gas, which also gives\nrise to Mg~{\\sc i} and Fe~{\\sc ii} transitions.\nGenerally, these profiles appear to exhibit ``characteristics''\nrelated to the location and structure probed by the QSO line of sight.\nIn particular, we note the complex high velocity spread in the\n$z=0.51$ galaxy toward Q1254 and the $z=0.92$ galaxy toward Q1206.\nOne could interpret the optically--thick components of these profiles\nas arising from the disks of these galaxies.\nThe HVC--like optically--thin components are more difficult to\nunderstand.\nHowever, they are highly suggestive of a picture in which material in\ngalaxy halos is comprised of kinematically and physically distinct\nclumps, consistent with the Searle--Zinn (1978) picture of halo formation.\n\n\nThere is a similarity between the features of the $z=1.17$ profile\ntoward Q1421 and that of the optically--thick components of Q1206.\nThese two QALs may arise in similar structures, or parts of\nthe galaxies.\nThe $z=1.55$ system toward Q1213 may be a merging or double galaxy,\nthe strength variations in the ``double'' profile being due to\nthe different line--of--sight orientations, morphologies, and\/or\nmasses of the galaxies.\nHST imaging would be decisive in testing these conjectures.\nSuch inferences can be drawn from QAL studies of local galaxies.\nBowen, Blades, \\& Pettini (1995) have observed a ``double'' profile\nsimilar to that of Q1213 that samples a line of sight passing\nthrough both M81 and the Milky Way and spans 400~km~s$^{-1}$.\nChurchill, Vogt, \\& Steidel (1995) have observed a possible double\ngalaxy at $z=0.74$ that exhibits a richly structured ``double''\nprofile spanning 300~km~s$^{-1}$.\nIn a ground--based image, there are two galaxies of nearly equal\nmagnitude (redshift?), each with impact parameter $\\sim 20$~kpc.\n\nIn Fig.~2, we present the probability, $P(\\Delta v)$, of observing any\ntwo clouds with line--of--sight velocity difference $\\Delta v$.\nThe cloud--cloud velocity dispersion within halos exhibits strong\nredshift evolution, becoming tighter as redshift decreases such that\nby a look--back time of $\\sim$ 8--10$h^{-1}$~Gyr\n($h=H_0\/100$~km~s$^{-1}$ Mpc$^{-1}$) Mg~{\\sc ii} absorbing clouds have\na mean velocity dispersion $\\sim 60$~km~s$^{-1}$.\nSuch {\\it pronounced}\\\/ evolution may be due to biasing in\nour sample selection, since we targeted an absorber population known\nto exhibit evolution (Steidel \\& Sargent 1992).\nSince there is no apparent evolution in the sizes of halos (Steidel \\&\nSargent 1992), or the number of clouds (this study), the implications\nare that an observable shift may be occurring in the mechanisms giving\nrise to significant amounts of high velocity material as early as\n$\\sim 8$ billion years ago.\nPerhaps we are seeing the epoch at which the frequency of dwarf\nsatellite galaxy accretion onto their primary slows considerably. \n\n\\acknowledgments\nWe would like to thank the Organizing Committee for hosting an\nenjoyable meeting and a small travel grant for CWC.\nThanks to Jane Charlton, Ken Lanzetta, and Chuck Steidel for \ninsightful on--going discussions.\nThis work has been supported in part by the Sigma Xi Grants--in--Aid of\nResearch program, the California Space Institute, and NASA\ngrant NAGW--3571.\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\par\nThe $125~ {\\rm GeV}$ Higgs boson was found by the ATLAS and CMS collaborations in 2012, which pastes the last brick of the Standard Model (SM). While physicists were celebrating the success of the SM, several problems emerged in front of us. Definitely we all know by bottom of hearts that the SM by no means is the final theory of the Nature but only an effective one at the concerned energy scale. Its loopholes, such as the naturalness of Higgs boson and the vacuum stability, all compose serious challenge to our knowledge. Moreover, some experimental phenomena show slight deviations from the SM predictions which reveal traces of new physics beyond SM (BSM). Indeed, the target of high energy society is searching for BSM and it especially is the task of LHC which historically succeeded in the SM Higgs boson discovery. Among all possible signals of BSM, the most favorable and significant signal of BSM is the existence of a new Higgs-like boson(s) which is predicted by many new models about BSM.\n\n\\par\nRecently, a preliminary investigation based on the report by S. Durgut, which is available at the American Physical Society (APS) site \\cite{Durgut}, shows a structure around $18.4~ {\\rm GeV}$ in four-lepton final state by using the four-lepton events collected during the LHC Run I stage. Below, we just refer the report as D-report. After carefully analysis, they conclude that an enhancement exists at around $18.4~ {\\rm GeV}$ in the invariant mass distribution of $\\Upsilon l^+l^- ~ (l = e,\\, \\mu)$ \\cite{Durgut,Yi:2018fxo,phdpaper}.\n\n\\par\nThe enhancement is conjectured to occur through the process $pp \\rightarrow \\Upsilon l^+ l^- \\rightarrow \\mu^+\\mu^-l^+l^-$. Namely if $\\Upsilon l^+l^-$ comes from a unique enhancement, it would be a neutral boson $\\phi$, which is mainly produced via gluon-gluon fusion, and then sequently decays as $\\Upsilon\\Upsilon^* \\rightarrow \\mu^+\\mu^- l^+l^-$, where $\\Upsilon^*$ might be off-mass-shell due to the energy constraint. The mass of the new enhancement, if it indeed exists, is a few hundreds of MeV lower than the total mass of a $\\Upsilon$ pair. By analyzing the datasets collected by the CMS detector at a center-of-mass energy of $7~ {\\rm TeV}$ and $8~ {\\rm TeV}$ with an integrated luminosity of $25.6~ {\\rm fb}^{-1}$ \\cite{phdpaper}, and taking the kinematic requirements of $p_{T,l} > 2.0~ {\\rm GeV}$ and $\\left| \\eta_{l} \\right| < 2.4$ on the final-state leptons \\cite{Durgut,phdpaper}, the experimenters observed a peaking structure. By their rigorous analysis, the peak is located at $18.4 \\pm 0.1(stat.) \\pm 0.2(syst.)~ {\\rm GeV}$ for the $\\Upsilon \\mu^+\\mu^-$ and $\\Upsilon e^+e^-$ channels combined; event numbers are $44 \\pm 13$ and $35 \\pm 13$ for the $\\Upsilon \\mu^+\\mu^-$ and $\\Upsilon e^+e^-$ channels, respectively, and its significance is $3.6$ standard deviation after taking into account the look-elsewhere-effect \\cite{Durgut,phdpaper}. Moreover, the ANDY collaboration also claims that a significant peak at $m = 18.12 \\pm 0.15(stat.) \\pm 0.6(syst.)~ {\\rm GeV}$ in the dijet mass distribution is observed in Cu+Au collision at $\\sqrt{s_{NN}} = 200~{\\rm GeV}$ at the Relativistic Heavy Ion Collider \\cite{Bland:2019aha}. This result is in good agreement with the four-lepton signal observed at the $7~ {\\rm TeV}$ and $8~ {\\rm TeV}$ LHC by the CMS collaboration \\cite{Durgut,phdpaper}.\n\n\\par\nBecause this enhancement is close to the sum of the masses of four bottom quarks, thus some authors consider it to be a $bb\\bar{b}\\bar{b}$ tetraquark state with a mass in the range of $18.4~ {\\rm GeV} < m_{X_{bb\\bar{b}\\bar{b}}} < 20.3~ {\\rm GeV}$ \\cite{Berezhnoy:2011xn,Wu:2016vtq,Chen:2016jxd,Bai:2016int,Anwar:2017toa,Richard:2017vry,Wang:2017jtz}. The authors of Ref.\\cite{Karliner:2016zzc} show that $\\sigma(pp \\rightarrow X_{bb\\bar{b}\\bar{b}}[0^{++}] \\rightarrow 4l)\\leqslant 4~ {\\rm fb}$ at $\\sqrt{s} = 13~ {\\rm TeV}$ and $\\leqslant 2~ {\\rm fb}$ at $\\sqrt{s} = 8~ {\\rm TeV}$. In Ref.\\cite{Esposito:2018cwh} the authors emphasize that their work was motivated by the peak at $18.4~ {\\rm GeV}$ based on the tetraquark hypothesis, but the numerical result shows that the partial width for $X_{bb\\bar{b}\\bar{b}} \\rightarrow \\Upsilon\\mu^+\\mu^-$ is too small to tolerate the data currently observed at the LHC. In Ref.\\cite{Becchi:2020mjz} the authors calculate the decay width for $X_{bb\\bar b\\bar b} \\rightarrow \\Upsilon l^+ l^-$ and they prefer $X_{bb\\bar{b}\\bar{b}}$ to be a $2^{++}$ tetraquark state rather than a $0^{++}$ bound state. Furthermore, the authors of Ref.\\cite{Hughes:2017xie} believe that a $bb\\bar{b}\\bar{b}$ tetraquark should lie above the lowest noninteracting bottomonium-pair threshold.\n\n\\par\nBecause the mass is just a bit below the sum of two $\\Upsilon$ bosons, being driven by the expectation of searching for a BSM Higgs-like boson, it is tempted to conjecture the newly observed enhancement to be a $0^{++}$ fundamental boson. In this work, our purpose is to check if the idea could be tolerated by the experimental observation. To serve this goal, we calculate the production rate of $\\Upsilon l^+l^-$ at the LHC by assuming the peak observed in experiment to be real, and then estimate the full contribution from the $18.4~ {\\rm GeV}$ structureless BSM boson $\\phi(18.4)$ as well as the corresponding SM background. In this paper, the $0^{++}$ enhancement $\\phi(18.4)$ is assumed as a BSM Higgs-like boson with mass around $18.4~ {\\rm GeV}$. It should be noted that if $\\phi(18.4)$ were indeed a Higgs-like boson, and its width is large enough, a threshold effect would induce an asymmetric peak in the invariant mass spectrum of $\\Upsilon l^+l^-$ which could be observed in experiments. Thus we estimate the possibility by numerically calculating the production of $\\Upsilon l^+l^-$. We eventually find that the production rate induced by the Higgs-like boson $\\phi(18.4)$ is too small to be observed in the LHC with presently available experimental condition. It means that if we deliberately postulate a large width for $\\phi(18.4)$, the contribution of the supposed BSM model may generate an experimentally observed peak around $18.4~ {\\rm GeV}$, however the data says no.\n\n\\par\nDiscovering new physics beyond SM should begin with looking for a new extra Higgs-like boson(s), this strategy is commonly accepted by both experimentalists and theorists in the high energy physics society. So far by now, many BSM models predict various kinds of new Higgs-like bosons (for example, neutral, charged, $\\mathcal{CP}$-odd or $\\mathcal{CP}$-even etc., even doubly-charged bosons). Unfortunately, none of them was found in present experiments so far. When looking back, we find that almost all the particles predicted by those BSM models are much heavier than the SM scale, namely it varies from few hundreds of GeV to few hundreds of TeV. Such BSM particles cannot be produced in present experimental facilities. On other aspect, there does not exist a principle forbidding the existence of lighter BSM particles. For example, the two-Higgs-doublet-model may predict a new Higgs-like boson with a mass of $28~ {\\rm GeV}$ \\cite{Cici:2019zir}. By the general method adopted for searching TeV-scale particles at LHC, alternatively, we, in this work, explore a new Higgs-like boson at low energy regions. As a common sense the strategy can be traced back from our experience gained at lepton colliders, such as BES, Belle, etc. For example, in the scattering process $e^+e^- \\rightarrow J\/\\psi \\rightarrow \\text{{\\it final products}}$, the resonance ($J\/\\psi$) overwhelmingly dominates the portal, while the direct production just provides a continuous background. For the same cause, a direct production of four leptons from the gluon-gluon fusion at the proton-proton collider, i.e., $gg \\rightarrow \\Upsilon \\Upsilon^* \\rightarrow \\mu^+\\mu^- l^+l^-$ \\cite{Li:2009ug,Qiao:2009kg} where $\\Upsilon^*$ might be off-mass-shell, should just generate a background. If a medium Higgs-like boson $\\phi(18.4)$ indeed exists, it induces the portal of $\\Upsilon \\Upsilon^* \\rightarrow \\Upsilon l^+l^-$, a peak would appear in the invariant mass spectrum of $\\Upsilon l^+l^-$. With this assertion, we numerically calculate the contribution induced by the BSM Higgs-like boson $\\phi(18.4)$ to $pp \\rightarrow \\Upsilon \\Upsilon^* \\rightarrow \\Upsilon l^+l^-$ at the LHC. Comparing our numerical results with those in the D-report \\cite{Durgut}, we find that the assumption that the observed peak in the $\\Upsilon l^+l^-$ mass spectrum originates from a BSM Higgs-like boson decay should be ruled out.\n\n\n\\par\nThis work is organized as follows. After this introduction, in section II, we present our analytical calculation for $\\Upsilon l^+l^-$ production at the LHC in the framework of a BSM model, in which we assume that the interaction of the BSM Higgs-like boson with SM particles is in analogue to that of the SM Higgs boson. In section III, we numerically evaluate all corresponding quantities and illustrate the invariant mass distribution of the final state. The last section is devoted to our conclusion and a brief discussion.\n\n\n\\section{Analytical calculation for $pp \\rightarrow \\Upsilon l^+l^-$}\nAt the LHC, $\\Upsilon l^+l^-$ is mainly produced via gluon-gluon fusion \\cite{Georgi:1977gs,Anastasiou:2002yz}, i.e.,\n\\begin{eqnarray}\n\\sigma[pp \\rightarrow \\Upsilon l^+l^-] = \\int dx_1 dx_2\\, f(x_1, \\mu_F)\\, f(x_2, \\mu_F)\\, \\hat{\\sigma}[gg \\rightarrow \\Upsilon l^+l^-],\n\\label{eqn1}\n\\end{eqnarray}\nwhere $f(x, \\mu_F)$ is the gluon distribution function in proton, $\\mu_F$ is the factorization scale, while other production channels are neglected.\n\n\\par\nThe contribution of the BSM Higgs-like boson comes from the Breit-Wigner propagator $\\dfrac{1}{p^2-m_{\\phi}^2 + i m_{\\phi} \\Gamma_{\\phi}}$, where $p$ is the four-momentum flowing through the intermediate BSM Higgs-like boson. If we do not consider the interference with the SM background and neglect the $t$- and\n$u$-channel Feynman diagrams induced by the BSM Higgs-like boson, this contribution will be proportional to the square of the Breit-Wigner propagator $\\dfrac{1}{( s - m_{\\phi}^2 )^2 + m_{\\phi}^2 \\Gamma_{\\phi}^2}$, where $s = (k_1+k_2)^2$ and $k_i~ (i = 1, 2)$ are the four-momenta of the two initial-state gluons. When $s$ is close to $m_{\\phi}^2$, this factor turns into $\\dfrac{1}{m_{\\phi}^2 \\Gamma_{\\phi}^2}$ and a resonance would peak up from the\nbackground. However, if $s$ is far away from $m_{\\phi}^2$ (below or above), the contribution of $\\phi$ would be drowned into the background and no peak can be seen. In our case, $18.4~ {\\rm GeV}$ is slightly below the threshold of $2 m_{\\Upsilon}$. However, since its position is not too far from the threshold value and it possesses a relatively large width, the resonance effect still can manifest itself in the invariant mass spectrum of $\\Upsilon$ pair at the threshold. In one aspect, the mass of $\\phi$ cannot be larger than $2 m_{\\Upsilon}$, otherwise a peak at the $\\Upsilon$ pair invariant mass spectrum would be seen, but no such peak was experimentally observed.\n\n\\par\nTo evaluate the contribution of the supposed BSM Higgs-like boson $\\phi$ of $18.4~ {\\rm GeV}$ to the $\\Upsilon l^+l^-$ production at the LHC, we write up the complete expression where the Breit-Wigner propagator of $\\phi$ with a width observed in the concerned experiment would induce the peak in the $\\Upsilon l^+l^-$ invariant mass spectrum. Later our numerical results show that one only needs to account the contribution of the resonance above the threshold of $2 m_{\\Upsilon}$. Indeed, because $18.4~ {\\rm GeV}$ is smaller than 2$m_{\\Upsilon}$, $\\phi$ cannot be on its mass-shell for two on-shell $\\Upsilon$s, while the production rate for $pp \\rightarrow \\phi \\rightarrow \\Upsilon\\Upsilon^* \\rightarrow \\Upsilon l^+l^-$ is very tiny and can be neglected. Due to the extremely small decay width of $\\Upsilon$ ($\\Gamma_{\\Upsilon} \\sim 50~ {\\rm keV}$ and $\\Gamma_{\\Upsilon} \\ll \\Gamma_{\\phi}$), the parton-level cross section for the production of $\\Upsilon l^+l^-$ via gluon-gluon fusion can be written as\n\\begin{eqnarray}\n\\hat{\\sigma}[gg \\rightarrow \\Upsilon l^+l^-] \\simeq \\hat{\\sigma}[gg \\rightarrow \\Upsilon \\Upsilon]\n\\times\n2 \\, Br(\\Upsilon \\rightarrow l^+l^-).\n\\end{eqnarray}\nThe cross section $\\hat{\\sigma}[gg \\rightarrow \\Upsilon \\Upsilon]$ is given by\n\\begin{eqnarray}\n\\hat{\\sigma}[gg \\rightarrow \\Upsilon \\Upsilon] = \\int d \\Omega \\, \\left| \\mathcal{M}_{SM} + \\mathcal{M}_{\\phi} \\right|^2,\n\\label{sigmahat}\n\\end{eqnarray}\nwhere $\\mathcal{M}_{SM}$ and $\\mathcal{M}_{\\phi}$ represent the Feynman amplitudes in the SM and induced by the BSM Higgs-like boson $\\phi$, respectively. It is noted that the above formula is a general expression where we do not specially require the intermediate boson $\\phi$, if it indeed exists in the nature, to be real or virtual. The production of $\\Upsilon$ pair via gluon-gluon fusion at hadron colliders has been much investigated in the framework of the SM \\cite{Li:2009ug,Qiao:2009kg}. The $31$ Feynman diagrams for $gg \\rightarrow \\Upsilon \\Upsilon$ in the SM can be created with the help of {\\sc FeynArts} \\cite{Hahn:2000kx} package. We also calculate this process with the same input parameters as given in Ref.\\cite{Li:2009ug} for comparison, and find that our numerical result for the production cross section at the $14~ {\\rm TeV}$ LHC is in good agreement with the corresponding one of Ref.\\cite{Li:2009ug} within a tolerable calculation error. Then we step on to calculate the quantities concerning the new Higgs-like boson.\n\n\\par\nAs for the BSM contribution from the new Higgs-like boson $\\phi$, we consider only the $gg\\phi$ and $b\\bar{b}\\phi$ effective couplings for the Higgs-like boson. The Feynman diagrams induced by the BSM Higgs-like boson $\\phi$ can be classified into two categories. The diagrams in Fig.\\ref{fig1} are independent of the $b\\bar{b}\\phi$ coupling, while all the diagrams in Fig.\\ref{fig2} depend on both $gg\\phi$ and $b\\bar{b}\\phi$ couplings.\n\\begin{figure}[htbp]\n \\centering\n \\subfigure[]{\n \\includegraphics[scale=0.35]{fig1a.eps}}\n \\subfigure[]{\n \\includegraphics[scale=0.35]{fig1b.eps}} \\\\\n \\subfigure[]{\n \\includegraphics[scale=0.35]{fig1c.eps}}\n\\caption{Feynman diagrams for $gg \\rightarrow \\Upsilon\\Upsilon$ induced by the BSM Higgs-like boson $\\phi$ via the $gg\\phi$ effective coupling.}\n\\label{fig1}\n\\end{figure}\n\\begin{figure}[htbp]\n \\centering\n \\subfigure[]{\n \\includegraphics[scale=0.35]{fig2a.eps}}\n \\subfigure[]{\n \\includegraphics[scale=0.35]{fig2b.eps}} \\\\\n \\subfigure[]{\n \\includegraphics[scale=0.35]{fig2c.eps}}\n \\subfigure[]{\n \\includegraphics[scale=0.35]{fig2d.eps}}\n\\caption{Feynman diagrams for $gg \\rightarrow \\Upsilon\\Upsilon$ induced by the BSM Higgs-like boson $\\phi$ via both $gg\\phi$ and $b\\bar{b}\\phi$ effective couplings.}\n\\label{fig2}\n\\end{figure}\n\n\\par\nFollowing Refs.\\cite{Anastasiou:2015ema,Spira:2016ztx}, the effective coupling of the BSM Higgs-like boson to two gluons can be written as\n\\begin{eqnarray}\n\\label{VggH}\n\\mathcal{C}_{gg\\phi}^{\\mu \\nu}(k_1,\\, k_2)\n=\n-i\\, \\dfrac{g_{gg\\phi}(\\mu_R)}{m_{\\phi}} \\left[ 4k_1 \\cdot k_2 \\Big( g^{\\mu\\nu}-\\frac{k_1^\\nu k_2^\\mu}{k_1\\cdot k_2} \\Big) \\right],\n\\end{eqnarray}\nwhere $k_1,~ k_2$ and $\\mu,~ \\nu$ are the four-momenta and Lorentz indices of the two gluons, respectively, $g_{gg\\phi}(\\mu_R)$ is a dimensionless effective running coupling constant, and $\\mu_R$ is the renormalization scale. It is reasonable to assume that the evolution of the effective coupling constant $g_{gg\\phi}$ is the same as that of the QCD $\\alpha_s$, i.e., $\\dfrac{g_{gg\\phi}(\\mu_R)}{\\alpha_s(\\mu_R)}$ is independent of $\\mu_R$. Thus, we obtain the quark-level amplitude for the Feynman diagrams in Fig.\\ref{fig1} as\n\\begin{eqnarray}\n\\widetilde{\\mathcal{M}}_{\\phi}^{(g)}\n=\n-i \\frac{4 \\pi \\alpha_s(\\mu_R) \\lambda_{{\\rm color}}}{(p_1 + q_2)^2 (p_2 + q_1)^2}\\,\n\\epsilon_\\mu(k_1) \\epsilon_\\nu(k_2)\\,\n{\\rm Tr} \\Big[ v(p_2) \\bar{u}(p_1) \\gamma_{\\alpha} v(q_2) \\bar{u}(q_1) \\gamma_{\\beta} \\Big]\n\\Big(\n\\mathcal{S} + \\dfrac{\\mathcal{T} + \\mathcal{U}}{8}\n\\Big),\n\\end{eqnarray}\nwhere $\\lambda_{{\\rm color}} \\equiv \\delta^{ab} {\\rm Tr}(T^aT^b) = 4$, and $\\mathcal{S}$, $\\mathcal{T}$ and $\\mathcal{U}$ are given by\n\\begin{eqnarray}\n&&\n\\mathcal{S} =\n\\mathcal{C}^{\\mu\\nu}_{gg\\phi}(k_1,\\, k_2)\\,\n\\mathcal{C}^{\\alpha\\beta}_{gg\\phi}(p_1+q_2,\\, p_2+q_1)\n\\left\/\n\\Big[ (k_1 + k_2)^2 - m_{\\phi}^2 + i m_{\\phi} \\Gamma_{\\phi} \\Big]\n\\right.,\n \\\\\n&&\n\\mathcal{T} =\n\\mathcal{C}^{\\mu\\alpha}_{gg\\phi}(k_1,\\, p_1+q_2)\\,\n\\mathcal{C}^{\\nu\\beta}_{gg\\phi}(k_2,\\, p_2+q_1)\n\\left\/\n\\Big[ (p_1 + q_2 - k_1)^2 - m_{\\phi}^2 + i m_{\\phi} \\Gamma_{\\phi} \\Big]\n\\right.,\n \\\\\n&&\n\\mathcal{U} =\n\\mathcal{T}\\,\\Big|_{k_1 \\leftrightarrow k_2,~ \\mu \\leftrightarrow \\nu}\\,.\n\\end{eqnarray}\nThe effective coupling of the BSM Higgs-like boson to the bottom quarks is parameterized as\n\\begin{eqnarray}\n\\label{VbbH}\n\\mathcal{C}_{b\\bar{b}\\phi} = -i\\, g_{b\\bar{b}\\phi}(\\mu_R).\n\\end{eqnarray}\nWe assume that the evolution of the effective coupling constant $g_{b\\bar{b}\\phi}$ is the same as that of the bottom-quark $\\overline{{\\rm MS}}$ running mass $\\overline{m}_b(\\mu_R)$ \\cite{Bednyakov:2016onn,Tanabashi:2018oca}, i.e., $\\dfrac{g_{b\\bar{b}\\phi}(\\mu_R)}{\\overline{m}_b(\\mu_R)}$ is independent of $\\mu_R$. Then the quark-level amplitude for the Feynman diagrams in Fig.\\ref{fig2} can be expressed as\n\\begin{eqnarray}\n\\widetilde{\\mathcal{M}}_{\\phi}^{(b)}\n=\ni \\frac{4 \\pi \\alpha_s(\\mu_R) \\lambda_{{\\rm color}}}{(k_1 + k_2)^2 - m_{\\phi}^2 + i m_{\\phi} \\Gamma_{\\phi}}\\,\n\\epsilon_\\mu(k_1) \\epsilon_\\nu(k_2)\\,\n\\mathcal{C}^{\\mu\\nu}_{gg\\phi}(k_1,\\, k_2)\\,\n\\mathcal{C}_{b\\bar{b}\\phi}\n\\Big(\n\\mathcal{F}_a + \\mathcal{F}_b + \\mathcal{F}_c + \\mathcal{F}_d\n\\Big),\n\\end{eqnarray}\nwhere $\\mathcal{F}_a$, $\\mathcal{F}_b$, $\\mathcal{F}_c$ and $\\mathcal{F}_d$ are given by\n\\begin{eqnarray}\n&&\n\\mathcal{F}_a\n=\n\\dfrac{1}{(p_2 + q_1)^2}\\,\n{\\rm Tr}\n\\biggl[\nv(p_2) \\bar{u}(p_1) \\gamma^{\\alpha}\n\\dfrac{1}{(\\slashed{k}_1 + \\slashed{k}_2) - \\slashed{q}_2 - m_b}\nv(q_2) \\bar{u}(q_1) \\gamma_{\\alpha}\n\\biggr],\n\\\\\n&&\n\\mathcal{F}_b\n=\n\\dfrac{1}{(p_2 + q_1)^2}\\,\n{\\rm Tr}\n\\biggl[\nv(p_2) \\bar{u}(p_1)\n\\dfrac{1}{\\slashed{p}_1 - (\\slashed{k}_1 + \\slashed{k}_2) - m_b} \\gamma^{\\alpha}\nv(q_2) \\bar{u}(q_1) \\gamma_{\\alpha}\n\\biggr],\n\\\\\n&&\n\\mathcal{F}_c\n=\n\\mathcal{F}_b\\,\\Big|_{p_1 \\leftrightarrow q_1,~ p_2 \\leftrightarrow q_2}\\,,\n\\\\\n&&\n\\mathcal{F}_d\n=\n\\mathcal{F}_a\\,\\Big|_{p_1 \\leftrightarrow q_1,~ p_2 \\leftrightarrow q_2}\\,.\n\\end{eqnarray}\nWithin the framework of NRQCD \\cite{Hao:2006nf}, the hadron-level amplitudes $\\mathcal{M}_{\\phi}^{(g)}$ and $\\mathcal{M}_{\\phi}^{(b)}$ can be obtained from the quark-level amplitudes $\\widetilde{\\mathcal{M}}_{\\phi}^{(g)}$ and $\\widetilde{\\mathcal{M}}_{\\phi}^{(b)}$, respectively, by performing the following replacement:\n\\begin{eqnarray}\n&&\nv(p_2)\\bar{u}(p_1)\n~~\\longrightarrow~~\n\\dfrac{1}{2\\sqrt{2}} \\, \\rlap\/\\epsilon^*_{\\Upsilon} \\big( \\rlap\/p + m_{\\Upsilon} \\big)\n\\dfrac{1}{\\sqrt{m_b}}\n\\Psi_{\\Upsilon}(0) \\dfrac{1}{\\sqrt{N_c}}\n\\\\\n&&\nv(q_2)\\bar{u}(q_1)\n~~\\longrightarrow~~\n\\dfrac{1}{2\\sqrt{2}} \\, \\rlap\/\\epsilon^*_{\\Upsilon} \\big( \\rlap\/q + m_{\\Upsilon} \\big)\n\\dfrac{1}{\\sqrt{m_b}}\n\\Psi_{\\Upsilon}(0) \\dfrac{1}{\\sqrt{N_c}}\n\\\\\n&&\np_1 = p_2 = \\dfrac{p}{2}\n\\\\\n&&\nq_1 = q_2 = \\dfrac{q}{2}\n\\end{eqnarray}\n\n\\par\nThe analytic expression of the SM amplitude for $gg \\rightarrow \\Upsilon\\Upsilon$ (i.e., $\\mathcal{M}_{SM}$) can be obtained analogously, but is not presented here since it is too tedious. Through the standard manipulations, we obtain the cross section $\\hat{\\sigma}[gg \\rightarrow \\Upsilon\\Upsilon]$ (Eq.(\\ref{sigmahat})), and then a convolution with the gluon distribution function results in the cross section for $pp \\rightarrow \\Upsilon\\Upsilon$. In next section we will show our numerical results clearly.\n\n\n\\section{Numerical results}\n\\par\nS. Durgut reported a peak in the invariant mass distribution of $\\Upsilon l^+l^-$ at the energy of $18.4~ {\\rm GeV}$ \\cite{Durgut,phdpaper}. A naive conjecture suggests that the peak at $M_{\\Upsilon l^+l^-} \\sim 18.4~ {\\rm GeV}$ is induced by a BSM Higgs-like boson. Our goal is to check if this scenario works. In this work, the event samples are generated by using {\\sc FormCalc} \\cite{Hahn:2016ebn} package based on the Monte Carlo technique. In the numerical calculation, the mass of the BSM Higgs-like boson is set as $18.4~ {\\rm GeV}$, and thus we denote this Higgs-like boson as $\\phi(18.4)$. The factorization and renormalization scales are set to the transverse energy of the final-state $\\Upsilon$, i.e., $\\mu_F = \\mu_R = \\sqrt{m_{\\Upsilon}^2 + p_{T,\\Upsilon}^{\\,2}}$. The masses of $b$-quark and $\\Upsilon$ are taken as $m_b = 4.73~ {\\rm GeV}$ and $m_{\\Upsilon} = 9.46~ {\\rm GeV}$ \\cite{Tanabashi:2018oca}. Within the framework of NRQCD, the zero point wave function of $\\Upsilon$ and the branching ratios of $\\Upsilon$ to $\\mu^+\\mu^-$ and $e^+e^-$ are taken as $\\Psi_{\\Upsilon}^2(0) = 0.391~ {\\rm GeV}^3$ \\cite{Li:2009ug,Quigg:1977dd,Eichten:1994gt,Eichten:1995ch}, $Br(\\Upsilon \\rightarrow \\mu^+\\mu^-) = 2.48\\%$ and $Br(\\Upsilon \\rightarrow e^+e^-) = 2.38\\%$ \\cite{Tanabashi:2018oca}. The gluon distribution function and the strong coupling constant $\\alpha_s$ are adopted from {\\sc CT14LO} \\cite{Schmidt:2015zda}.\n\n\n\\par\nThe dependence of the cross section $\\sigma_{\\phi}[pp \\rightarrow \\Upsilon\\mu^+\\mu^-]$, defined by $\\big| \\mathcal{M}_{\\phi} = \\mathcal{M}_{\\phi}^{(g)} + \\mathcal{M}_{\\phi}^{(b)} \\big|^2$, on the effective coupling constants $g_{gg\\phi}$ and $g_{b\\bar{b}\\phi}$ at the $8~ {\\rm TeV}$ LHC is shown in Fig.\\ref{fig3}. Different colors of the points in Fig.\\ref{fig3} represent different values of $\\sigma_{\\phi}$. The parameter space region above the red line is excluded by the experimental constraint from the decay width of $\\phi(18.4)$, i.e.,\n\\begin{eqnarray}\n\\Gamma[\\phi(18.4) \\rightarrow gg]+ \\Gamma[\\phi(18.4) \\rightarrow b\\bar b] < \\Gamma[\\phi(18.4) \\rightarrow all] \\simeq 35~ {\\rm MeV}.\n\\end{eqnarray}\nIn the experimentally allowed region of the parameter space, the signal cross section $\\sigma_{\\phi}$ reaches its maximum at the parameter point ${\\rm A} = (0.0344,\\, 0.0829)$,\n\\begin{eqnarray}\n\\sigma_{\\phi}[ pp \\rightarrow \\Upsilon\\mu^+\\mu^- \\, @ ~ 8~ {\\rm TeV} ]\\Big|_{{\\rm A}}\n=\n0.853~ {\\rm fb}.\n\\end{eqnarray}\nIt is obvious that $\\sigma_{\\phi}[ pp \\rightarrow \\Upsilon\\mu^+\\mu^- \\, @ ~ 8~ {\\rm TeV} ] < 0.853~ {\\rm fb}$ in the whole experimentally allowed parameter space region. The purpose of this study is to investigate whether the existence of a BSM Higgs-like boson can fit the enhancement observed by the CMS collaboration. Therefore, we give preference to the parameter point A in the following discussion.\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{fig3.eps}\n\\caption{Dependence of $\\sigma_{\\phi}[pp \\rightarrow \\Upsilon \\mu^+\\mu^-]$ on the effective coupling constants $g_{gg\\phi}$ and $g_{b\\bar{b}\\phi}$ at the $8~ {\\rm TeV}$ LHC. The parameter space region above the red line is excluded by the experimental constraint of $\\Gamma[\\phi(18.4) \\rightarrow gg]+ \\Gamma[\\phi(18.4) \\rightarrow b\\bar b] < \\Gamma[\\phi(18.4) \\rightarrow all] \\simeq 35~ {\\rm MeV}$.}\n\\label{fig3}\n\\end{center}\n\\end{figure}\n\n\\par\nThe integrated cross sections and invariant mass spectra of $\\Upsilon \\mu^+\\mu^-$ for $pp \\rightarrow \\Upsilon \\mu^+\\mu^-$ at the $8$ and $13~ {\\rm TeV}$ LHC are provided in Tab.\\ref{tab1} and Fig.\\ref{fig4}, respectively. The contributions from $\\left| \\mathcal{M}_{SM} \\right|^2$, $\\left| \\mathcal{M}_{\\phi} \\right|^2 + 2 {\\rm Re}\\big( \\mathcal{M}_{SM}^{\\dag} \\mathcal{M}_{\\phi} \\big)$ and $\\left| \\mathcal{M}_{\\phi} \\right|^2$, which are regarded as the SM background and the new physics signals induced by the BSM Higgs-like boson with and without interference effect, are provided separately, and labeled with $B$, $S$ and $\\hat{S}$ respectively. Table \\ref{tab1} clearly shows that the interference between the BSM amplitude induced by $\\phi(18.4)$ and the SM amplitude for $pp \\rightarrow \\Upsilon \\mu^+\\mu^-$ is negative, and thus reduces the new physics signal induced by $\\phi(18.4)$ in $pp \\rightarrow \\Upsilon \\mu^+\\mu^-$ production. One can notice that the contribution of $\\phi(18.4)$ at colliding energy between two gluons being below $2 m_{\\Upsilon}$ is almost zero, but would jump up at $\\sqrt{s} = 2 m_{\\Upsilon}$. It is a standard threshold effect. One characteristic of the phenomenon is the observed ``peak\" is not in the symmetric Gaussian form. Anyhow, even though we suppose existence of a BSM Higgs-like boson which may decay into $\\Upsilon\\mu^+\\mu^-$, it is impossible to induce a peak at $18.4~ {\\rm GeV}$ at all. What's more, the extremely narrow peak at $M_{\\Upsilon \\mu^+\\mu^-} \\sim 18.4~ {\\rm GeV}$ in the invariant mass spectrum of $\\Upsilon \\mu^+\\mu^-$ in the D-report ($\\Gamma[\\phi(18.4) \\rightarrow all] < 35~ {\\rm MeV}$) \\cite{Durgut,phdpaper} gives a stringent constraint on the effective couplings of the Higgs-like boson $\\phi(18.4)$, especially the coupling to two gluons.\n\\begin{table}[h]\n\\begin{center}\n\\renewcommand\\arraystretch{1.8}\n\\begin{tabular}{cccc}\n\\hline\n\\hline\n~~$\\sqrt{s}$~ [TeV]~~ & ~~$\\sigma_{\\hat{S}}$~ [fb]~~ & ~~$\\sigma_{S}$~ [fb]~~ & ~~$\\sigma_{B}$~ [pb]~~ \\\\\n\\hline\n$8$ & $0.853$ & $-0.637$ & $1.041$ \\\\\n$13$ & $1.464$ & $-1.073$ & $1.811$ \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{Integrated cross sections for $pp \\rightarrow \\Upsilon\\mu^+\\mu^-$ at the $8$ and $13~ {\\rm TeV}$ LHC. $B$ stands for the SM background, $S$ and $\\hat{S}$ represent the new physics signals induced by $\\phi(18.4)$ with and without interference effect, respectively. The decay width and the effective coupling constants of $\\phi(18.4)$ are taken as $\\Gamma_{\\phi} = 35~ {\\rm MeV}$, $g_{gg\\phi}(m_Z) = 0.0344$ and $g_{b\\bar{b}\\phi}(m_Z) = 0.0829$.}\n\\label{tab1}\n\\end{center}\n\\end{table}\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{fig4a.eps}\n\\includegraphics[width=0.6\\textwidth]{fig4b.eps}\n\\caption{$\\Upsilon\\mu^+\\mu^-$ invariant mass distributions for $pp \\rightarrow \\Upsilon\\mu^+\\mu^-$ at the $8$ and $13~ {\\rm TeV}$ LHC. $B$ stands for the SM background and $\\hat{S}$ represents the new physics signal induced by $\\phi(18.4)$ without interference effect, respectively. The decay width and the effective coupling constants of $\\phi(18.4)$ are taken as $\\Gamma_{\\phi} = 35~ {\\rm MeV}$, $g_{gg\\phi}(m_Z) = 0.0344$ and $g_{b\\bar{b}\\phi}(m_Z) = 0.0829$.}\n\\label{fig4}\n\\end{center}\n\\end{figure}\n\n\\par\nIn Tab.\\ref{tab2}, we present the production cross sections for both signal and SM background for $\\Upsilon e^+e^-$ final state. We can see clearly that the numerical results for the $pp \\rightarrow \\Upsilon e^+e^-$ channel are almost the same as those for $pp \\rightarrow \\Upsilon \\mu^+\\mu^-$ due to the lepton universality ($Br(\\Upsilon \\rightarrow e^+e^-) \\simeq Br(\\Upsilon \\rightarrow \\mu^+\\mu^-)$). Thus, we do not provide the invariant mass distribution of $\\Upsilon e^+e^-$ in this section.\n\\begin{table}[h]\n\\begin{center}\n\\renewcommand\\arraystretch{1.8}\n\\begin{tabular}{cccc}\n\\hline\n\\hline\n~~$\\sqrt{s}$~ [TeV]~~ & ~~$\\sigma_{\\hat{S}}$~ [fb]~~ & ~~$\\sigma_{S}$~ [fb]~~ & ~~$\\sigma_{B}$~ [pb]~~ \\\\\n\\hline\n$8$ & $0.819$ & $-0.611$ & $0.999$ \\\\\n$13$ & $1.405$ & $-1.030$ & $1.738$ \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{The same as Tab.\\ref{tab1} but for $\\Upsilon e^+e^-$ final state.}\n\\label{tab2}\n\\end{center}\n\\end{table}\n\n\n\\section{Discussions and conclusion}\n\\par\nBased on the data of the Run I of LHC at $7$ and $8~ {\\rm TeV}$, we investigate the origin of the peak at $18.4~ {\\rm GeV}$ in the invariant mass spectrum of $\\Upsilon l^+l^-$ newly reported in Refs.\\cite{Durgut,phdpaper}. We postulate it to be a $0^{++}$ BSM Higgs-like boson, and by the anzatz calculate the production rate of $\\Upsilon l^+l^-$ via gluon-gluon fusion at the LHC and discuss the effect of this BSM Higgs-like boson. In our calculations, we assume the effective couplings of the $0^{++}$ BSM Higgs-like boson to the gluons and bottom quarks have the same evolution behaviour as the corresponding ones of the SM Higgs boson.\n\n\\par\nFor the peak observed in the invariant mass spectrum of $\\Upsilon l^+l^-$ at $18.4~ {\\rm GeV}$ whose width was not accurately fixed yet, the situation might imply that the peak corresponds to a BSM Higgs-like boson which decays into $\\Upsilon\\Upsilon^*$ and later turns into $\\Upsilon l^+l^-$ and eventually goes to the four-lepton final state. The peak position is located at $18.4~ {\\rm GeV}$ which is lower than the threshold value of $2 m_{\\Upsilon}$, so that it impossibly directly decays into a real $\\Upsilon$ pair if we do not consider its width. If it possesses a relatively large width whose edge covers the region of $2 m_{\\Upsilon}$, it may result in an asymmetric peak at $M_{\\Upsilon l^+l^-} \\simeq 2 m_{\\Upsilon}$ in the invariant mass spectrum of $\\Upsilon l^+l^-$ via the threshold effect. We carefully analyze the possibility and our numerical results (Fig.\\ref{fig4}) assure that there cannot exist an even-not-very apparent asymmetric peak above $2 m_{\\Upsilon}$. Moreover, the contribution of the new Higgs-like boson to the portal $\\Upsilon l^+l^-$ would interfere with the SM contribution and accurate measurements may detect the variation. But all the numerical results do not manifest an appearance of a peak at $18.4~ {\\rm GeV}$.\n\n\\par\nFrom Tab.\\ref{tab1}, Tab.\\ref{tab2} and Fig.\\ref{fig4} we can notice that if the coupling constants are small and the width of the supposed Higgs-like boson is narrow ($\\Gamma[\\phi(18.4) \\rightarrow all] < 35~ {\\rm MeV}$), the cross section is $\\mathcal{O}(1~{\\rm fb})$, such a small cross section cannot be experimentally observed by the present facilities. Since the effect of the BSM Higgs-like boson $\\phi(18.4)$ cannot be detected in $\\Upsilon l^+l^-$ final state at the parameter point ${\\rm A} = (0.0344,\\, 0.0829)$, the whole parameter space region allowed by the constraint of $\\Gamma[\\phi(18.4) \\rightarrow gg]+ \\Gamma[\\phi(18.4) \\rightarrow b\\bar b] < \\Gamma[\\phi(18.4) \\rightarrow all] \\simeq 35~ {\\rm MeV}$ is entirely excluded by the peak observed in the $\\Upsilon l^+l^-$ invariant mass spectrum at $18.4~ {\\rm GeV}$. If this scenario is true, we would conclude that the peak at $18.4~{\\rm GeV}$ does not correspond to a $0^{++}$ BSM Higgs-like boson, but something else.\n\n\\par\nAll of our estimates are based on the experimental results reported in Refs.\\cite{Durgut,phdpaper}. Our numerical results decide that the peak may not corresponds to a $0^{++}$ BSM Higgs-like boson ($18.4~ {\\rm GeV}$). Definitely much more accurate measurements which will be carried out at future high energy facilities (including the updated LHC) will give more information about this peak.\n\n\\par\nBy our assumption, the observed peak is a BSM Higgs-like boson, if it is true, it would set a scale for the BSM and the significance is obvious. Indeed, for the peak appearing at the invariant mass spectrum of $\\Upsilon l^+l^-$, Refs.\\cite{Karliner:2016zzc,Esposito:2018cwh,Becchi:2020mjz} consider it to be a composite of $bb\\bar{b}\\bar{b}$, but all their study show that the decay width are too small to be currently observed at the LHC. The observation is important and following the data, the theoretical interpretation can be made. Since it implies new understanding on new physics beyond the SM and sets a new scale, obviously, the study along this line cannot be neglected. We hope the experimentalists of high energy physics to continue the investigation on the peak by more accurate measurement and analysis. The conclusion would greatly help theorists making a definite judgement to verify the validity of our ansatz or negate it.\n\n\\par\nNow let us make a brief summary and draw our conclusion (so far, but by no means for the future). In this work we are trying to investigate whether the enhancement observed at LHC is a structureless BSM boson. If it indeed is, it can contribute to the process of $pp \\rightarrow \\Upsilon l^+l^-$, but how it behaves, can it result in a peak at the invariant mass spectrum of $\\Upsilon l^+l^-$, in other words, does it induce the peak at $18.4~ {\\rm GeV}$ reported in Refs.\\cite{Durgut,phdpaper}? It demands a clear answer. Even though a BSM boson $\\phi$ exists and possesses a certain width, an inequality $m_{\\phi}+ \\Gamma_\\phi < 2 m_{\\Upsilon}$ holds. Our explicit computation indicates that $\\phi$ as an on-shell real particle may not directly contribute to $pp \\rightarrow \\phi \\rightarrow \\Upsilon\\Upsilon^* \\rightarrow \\Upsilon l^+l^-$. Thus even though a BSM Higgs-like boson $\\phi$ of $18.4~ {\\rm GeV}$ exists and may contribute to $pp \\rightarrow \\Upsilon l^+l^-$, the sizable rate only occurs above the threshold of 2$m_{\\Upsilon}$. But then $\\phi$ must be off-shell (or contributes via $t$- and $u$-channels), therefore our conclusion is that the experimentally observed peak located at $18.4~ {\\rm GeV}$ with a narrow width does not correspond to a BSM structureless Higgs-like boson. The peak of $18.4~ {\\rm GeV}$ must originate from other mechanism and its appearance cannot be a signature of existence of BSM as expected.\n\n\n\\vskip 10mm\n\\par\n\\noindent{\\large\\bf ACKNOWLEDGMENTS} \\\\\nThis work is supported in part by the National Natural Science Foundation of China (Grants No. 11675082, 11735010, 11775211, 11535002, 11805160, 11747040, 11375128, the Natural Science Foundation of Jiangsu Province of China (Grant No. BK20170247), Special Grant of the Xuzhou University of Technology (No. XKY2016211, XKY2017215, XKY2018221), and the CAS Center for Excellence in Particle Physics (CCEPP).\n\n\n\\vskip 5mm\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nPG~1002+506 was discovered by the Palomar-Green UV-excess survey (Green,\nSchmidt, \\& Liebert 1986) and listed as a cataclysmic variable (CV). \nDuring a study of the CVs from this survey, Ringwald (1993) obtained\nultraviolet and red spectra, and tentatively reclassified it as a detached\nsubdwarf binary, noting H$\\alpha$ in strong emission, unresolved at\n10-\\AA\\ resolution. Several puzzling aspects were noted, however,\nincluding the near-constancy of the radial velocities throughout two\nnights, consistent with no change other than that attributable to\natmospheric dispersion in an unrotated slit. There was also no\nsignificant variation in the equivalent width of H$\\alpha,$ which one\nmight expect if this were a detached CV progenitor with the hot component\nirradiating the facing hemisphere of its companion. \n\nThat PG~1002+506 is not a CV was shown definitively by E. L. Robinson\n(1995, private communication): it does not flicker, or have the erratic\nvariability ubiquitous in CVs. This was found with high-speed\nsimultaneous {\\it UBVR\\\/} photometry taken in 1995 June with the Stiening\nphotometer on the McDonald Observatory 2.1-m telescope. In 25~min of\nphotometry with 1-s time resolution, all bands showed peak-to-peak\namplitudes of $<\\,2$\\%. \n\nThis and further spectra have forced another reclassification of this\nstar, as a high-latitude Be star. This is one of two known in the\nPalomar-Green catalog, the other being PG~0914+001 (Saffer et al.~1997). \nAn Oe star from this survey is also known, PG~2120+062 (Moehler, Heber, \\&\nDreizler 1994). \n\nFor reviews on Be stars, see Jaschek \\& Jaschek (1987) and Slettebak\n(1988). About one in five non-supergiant B stars shows emission, mainly\nin H$\\alpha$ but sometimes also in H$\\beta$ and higher Balmer lines. \nStruve (1931) attributed this to a disk, extruded by the star's rotation\nnear the breakup velocity, $ \\sqrt{ G M \/ R}.$ What excites the emission\nin Be stars is a long-standing mystery, however, as is their evolutionary\nstatus. Although Be stars often have an IR excess, PG~1002+506 is not an\nIRAS source (Neugebauer et al.~1988). \n\n\n\\section{Blue spectrum}\n\nA blue spectrum (Figure 1) was taken in service time with the Intermediate\nDispersion Spectrograph on the Isaac Newton Telescope on La Palma. This\n1800-s spectrum was taken in photometric conditions in $2''$ seeing,\nthrough a $1.73''$~slit, and has 1.5-\\AA\\ (FWHM) resolution. The slit was\naligned to the parallactic angle, to avoid atmospheric dispersion effects;\nthe spectrum was taken when PG~1002+506 was nearly overhead, at an airmass\nof 1.08. \n\nA spectral classification of B$5\\pm 1$ V was arrived at by comparing this\nspectrum to model atmospheres (Kurucz 1979) and published spectra (Jacoby,\nHunter, \\& Christian 1984; Jaschek \\& Jaschek 1987). That this is a\nmain-sequence star and not a subdwarf is shown by the presence of the H13\nand 14 lines. That it is not a giant or supergiant is shown by the widths\nof its Balmer lines, with FWZI of H$\\gamma$ of $ 31 \\pm 3$~\\AA. There is\nno spectroscopic evidence that this star is a binary. \n\n\n\n\\section{Radial Velocity}\n\nOn 1997 January 3 UT, two 10-min exposures were obtained with the Modular\nSpectrograph on the 2.4-m Hiltner Telescope at Michigan-Dartmouth-MIT\nObservatory, Kitt Peak, Arizona. The spectra covered from 4650 to 6727\n\\AA, and had 4-\\AA\\ (FWHM) resolution. The weather was poor, with $>\n1\\arcsec$ seeing and rising humidity that forced a shutdown just after\nthese spectra were taken. The spectrograph slit was set at the\nparallactic angle, even though PG~1002+506 was only one hour east of the\nmeridian. The 1\\arcsec\\ slit projected to 3 \\AA\\ on the detector. With\nthe mediocre seeing, we expect ``slit-painting'' velocity errors to be\nsmall, probably $< 5$ km~s$^{-1},$ based on experience with similar sharp\nlines in white dwarf\/red dwarf binaries (Thorstensen, Vennes, \\& Shambrook\n1994). The exposures were bracketed by HgNeXe exposures, for which the\nRMS residual was $< 0.05$~\\AA, and the maximum residuals for the weakest\nlines were $< 10$~km~s$^{-1}.$ Most lines had residuals around\n2~km~s$^{-1}.$\n\nH$\\alpha$ appears to be slightly resolved, and is in strong emission (see\nFigure 2), with an equivalent width of $17.8 \\pm 0.3 $~\\AA\\ and FWHM of\n$580 \\pm 30$ km~s$^{-1}.$ There is also emission in the core of H$\\beta.$\nBy convolving H$\\alpha$ with the derivative of a Gaussian with FWHM = 8\n\\AA\\ and taking the zero of the convolution as the velocity (Schneider \\&\nYoung 1980), we find heliocentric radial velocities of the spectra taken\nat HJD 2450451.90425 and 2450451.91140 of +29.3 and +28.9~km~s$^{-1},$\nrespectively. The velocities of the O~I $\\lambda$\\,6300 \\AA\\ night sky\nline were 1.6 and 0.7~km~s$^{-1},$ showing the accuracy of the wavelength\nscale. \n\nHowever, the emission lines in Be stars are well known to be variable in\nprofile over timescales of days or longer, and are therefore not reliable\nindicators of the systemic velocity. The spectra were therefore summed\ntogether and rectified, to remove continuum slope effects. The radial\nvelocity was then measured from the absorption wings of H$\\alpha$ by\nconvolving a positive and a negative Gaussian with the line profile and\ntaking the zero of this convolution as the velocity (Schneider \\& Young\n1980). In all cases the Gaussians had 4 channels FWHM. The separation\nbetween the Gaussians was varied, from 24 to 20 to 16 \\AA; the\ncorresponding heliocentric radial velocities are $-2.0,$ $-0.5,$ and\n$-4.1$~km~s$^{-1}.$ Finding the line's centroid by fitting and subtracting\na linear approximation of the continuum, numerical integration of the\nintensity, and taking the centroid (crudely, with the IRAF {\\it splot\\\/}\n`e' command) gave +0.4 km~s$^{-1}.$ We conclude that PG~1002+506 has a\nheliocentric radial velocity of $-2 \\pm 15$~km~s$^{-1}.$\n\n\n\\section{Model atmosphere analysis}\n\nWe have performed a model atmosphere analysis of the blue optical spectrum\nto estimate the atmospheric parameters $T_{\\rm eff}$ and log $g$, as well\nas the projected stellar rotation velocity $v \\sin i$. Our grid of\nsynthetic spectra was calculated with the radiative transfer code SYNSPEC\n(Hubeny, Lanz, \\& Jeffrey 1995), assuming the temperature and pressure\nstratifications of Kurucz (1991). The metal and helium abundances were\nheld fixed at the solar value. At the temperature and surface gravity of\nspectral type B5V, the assumption of LTE is well justified. The\ntemperature and gravity grid points were $T_{\\rm eff}$ = 13,000 --\n17,000~K in steps of 1,000~K, and log $g$ = 3.5 -- 5.0 in steps of 0.5\ndex. In addition each model was convolved with a rotational broadening\nfunction at projected rotation velocities $v \\sin i$ = 50 -- 350\nkm~s$^{-1}$ in steps of 50 km~s$^{-1}$ to produce a 3-dimensional fitting\ngrid. The stellar parameters were estimated by simultaneous variation\nusing a non-linear $\\chi^2$ minimization algorithm. Details of the\nsynthetic spectrum calculations and the fitting algorithm are given by\nSaffer et al.~(1994) and Saffer et al.~(1997). Due to the partial filling\nin of the lower Balmer lines by emission from the circumstellar material,\nwe have restricted the analysis to the portion of the spectrum blueward of\nH$\\beta.$\n\nThe best-fit stellar parameters are $T_{\\rm eff} = 14,900 \\pm 1200$~K,\n$\\log g = 4.20 \\pm 0.2$, and $v \\sin i = 340 \\pm 50$ km~s$^{-1}$ (see\nFigure 1). The quoted 1-$\\sigma$ errors are based on counting statistics\nand account for covariance for the fitting parameters; they also estimate\nsystematic errors. \n\n\n\\section{Evolutionary status}\n\nThe effective temperature, surface gravity, and very high rotational\nvelocity are fully consistent with a spectral classification of B5Ve. The\nbreakup velocity expected for this star is 540 km~s$^{-1}.$ The fit places\nthis star in the area of confusion in the $T_{\\rm eff}\/\\log g$ diagram\nwhere the Population I main- sequence intersects the Population II blue\nhorizontal branch (BHB) (Sch\\\"onberner 1993; Bertelli et al.~1994). For\nexample, PG~0832+676 at first appeared to be a young star far from the\nGalactic plane, but turned out to be a nearby blue evolved star, upon\nanalysis of high-resolution spectra (Hambly et al.~1996). However,\nidentification of PG~1002+506 as a BHB star is contradicted both by the\nemission reversals in the H$\\alpha$ and H$\\beta$ absorption lines, and by\nits high rotation velocity, since BHB stars are slow rotators (Peterson,\nRood, \\& Crocker 1995). \n\nAssuming PG~1002+506 to be of Population I origin, we used the derived\natmospheric parameters and the evolutionary tracks of Claret \\& Gimenez\n(1992) to estimate the stellar mass and evolutionary age (see Table 1). A\ndistance estimate was obtained from the absolute visual magnitude deduced\nfrom the stellar mass, atmospheric parameters, and bolometric corrections\nof Kurucz (1979). PG~1002+506 has $B = 15.36$ (Green et al.~1986).\nAssuming $B - V = -0.16$ for B5V stars (Allen 1973), and a reddening\n$E(B-V) < 0.01$, inferred from the map of Burstein \\& Heiles (1982), this\nwould imply a distance of 13.9 kpc, which for a Galactic latitude $b =\n51^{\\circ},$ corresponds to a z-distance of 10.8 kpc above the Galactic\nplane. Although large, this is not unheard of (Kilkenny 1992). For a\nGalactic longitude $l = 165^{\\circ},$ this would imply a galactocentric\nradius of 17.1 kpc, putting PG~1002+506 at the outskirts of the Galaxy. \n\n\n\\section{Kinematical analysis}\n\nAs the existence of young objects at large distances from the star forming\nregions of the Galactic disk is potentially interesting, we have performed\na kinematical analysis for PG~1002+506. Although no proper motion\ninformation is available, it is possible to use the observed radial\nvelocity of a star to constrain its evolutionary history. A detailed\ndescription of the method of analysis is given by Rolleston et al.~(1997). \n\nWe first consider a scenario whereby PG~1002+506 has a zero velocity\ncomponent parallel to the Galactic disk, and ejection has occurred\nperpendicular to the plane of the Galaxy. We have corrected the observed\nheliocentric velocity for the effects of differential rotation (Fich et\nal.~1989), assuming that the halo co-rotates with the disk, to determine\nthe stellar radial motion $(v_r)$ with respect to a standard of rest\ndefined by its local environment. Our initial assumption implies that the\nobserved radial velocity is a component of the stellar space motion\n$(v_z)$ perpendicular to the disk. We then attempt to show that PG\n1002+506 could have reached its present position in the Galactic halo\nwithin its evolutionary lifetime, while reproducing the observed radial\nvelocity, and calculating the required ejection velocity. These\ncalculations have adopted the gravitational potential function of House \\&\nKilkenny (1980). This analysis implicitly assumes that the star is ejected\nfrom the disk shortly after birth, consistent with cluster ejection\nsimulations. \n\nThe results of the kinematical analysis are given in Table 1. Given the\nlarge z-distance, it is not surprising to find the ``time of flight'' to\nbe larger than the evolutionary age. We have therefore considered the\neffects of errors in the derived atmospheric parameters and the radial\nvelocity measurement. By optimizing the values of $T_{\\rm eff}$ and $\\log\ng$ such that they are self-consistent within the errors, it is possible to\nincrease the evolutionary age, so that it is greater than the predicted\nflight time. For example, adopting values of $T_{\\rm eff} = 13,750$~K and\n$\\log g = 4.0$ would imply an age of 115 Myr for a mass of 4.0\n$M_{\\odot}.$ Allowing an error of 15 km~s$^{-1}$ in the observed\nheliocentric velocity also decreases the estimated flight time, but not\nsignificantly, to 84 Myr. \n\n\n\\section{Conclusions}\n\nPG~1002+506 appears to be a young, rapidly rotating B5Ve star at a\ndistance of 10.8 kpc from the Galactic plane, and at a galactocentric\nradius of 17.1 kpc. The kinematical analysis suggests that it could have\nattained its present Galactic position having been ejected from the disk\nshortly after its formation. Furthermore, the required ejection velocity\nof $\\approx 230$~km~s$^{-1}$ can also be produced by the known mechanisms\npredicted by Leonard (1993). A detailed atmospheric analysis with\nhigher-quality spectra should still be done, to determine abundances and\nconfirm that PG~1002+506 really is a distant main-sequence star, and not a\nnearby blue evolved star. If PG~1002+506 really is 10.8 kpc from the\nGalactic plane, interstellar absorption in this same spectrum would probe\na line through the Galactic halo otherwise difficult to acquire. \n\n\n\\acknowledgments\\noindent\nE.~Harlaftis took the blue spectrum with the Isaac Newton telescope, which\nis operated on La Palma by the Royal Greenwich Observatory at the Spanish\nObservatorio del Roque de los Muchachos of the Instituto de Astrofisica de\nCanarias. Michigan-Dartmouth-MIT Observatory is operated by a consortium\nof the University of Michigan, Dartmouth College, and the Massachusetts\nInstitute of Technology. Thanks also to Rob Robinson, Malcolm Coe,\nRichard Green, Uli Heber, Gerrie Peters, and Richard Wade, for helpful\ndiscussions. \n\n\n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIntegrable models may have very huge symmetries, that \nhelp us to study various behaviors of the systems.\nFor some well investigated models, we are able to \ncalculate correlation functions of observables \nby using representation theories of the symmetries, that\nhave been fruitfully studied within the language of \ninfinite dimensional Lie algebras and their suitable\ndeformation theories. We now have a lot of\nexamples of these symmetry algebras and their applications \nto many problems.\nWhat is the natural candidate for that symmetry which is \npresent in majority of the integrable models?\nIn other words, what is the ``universal'' symmetry among all these?\n\n\nThe $1+1$-dimensional conformal field theories (CFT's) \n\\cite{rBPZ} describe the \nuniversality classes of massless field theories in $1+1$-dimensions.\nIt is the guideline given in the celebrated paper\n\\cite{rBPZ} \nthat all the CFT's must be \nregarded as representations of the ``Virasoro algebra''\nregardless of the details of the models.\nNamely, the Virasoro algebra is the universal one in CFT.\nIndeed, through the Sugawara construction \\cite{rKZ,rGO},\nthe Virasoro algebra exists in any Kac-Moody algebra.\nThese models are successfully \napplied to the critical phenomena of \ntwo-dimensional classical statistical models, \nthe low temperature behavior of one-dimensional electron systems and so on.\nIn any sense, CFT is definitely quite well understood among \nother interacting field theories, \nbecause we have the infinite dimensional conformal symmetry.\n\nWe are gradually understanding the universal symmetry arising \nin off-critical models: massive integrable field\ntheories in 1+1 dimension (sine-Gordon model etc.) \\cite{rL}, \none-dimensional quantum spin chain systems ($XYZ$ model etc.) \\cite{rL},\ntwo-dimensional solvable lattice \nmodels (ABF models etc.) \\cite{rLP1,rLP2,rAJMP},\ndeformations of the KdV hierarchy and \ndiscretized soliton equations \\cite{rFR,rFr},\nCalogero-Sutherland(CS)-type quantum mechanical \nmodels \\cite{rAMOS,rAOS} and so on.\nIt had been recognized that nontrivial\nVirasoro-type symmetries exist in these off-critical theories;\nin \\cite{rLP1}, the existence of two-parameter\nVirasoro-type symmetry was conjectured, and \na one-parameter Virasoro-type Poisson structure was\nfound in \\cite{rFR}.\nIt was in the CS-type quantum \nmechanical model, that \nwe finally obtained the definition and an \nexplicit construction of the two-parameter\nVirasoro-type symmetry which we call \\v \\cite{rSKAO}.\nThis is already extended to the case of ${\\cal W}$-algebra \\cite{rFFr,rAKOS}.\nIt is really astonishing that all the Virasoro-type algebras \nrelated with the off-critical integrable models are \nobtained by taking suitable limits from \\v.\nHowever,\nwe have not fully understand \nthe meaning of ``universality'' played by this new Virasoro-type\nsymmetry \\v, so far. \nThe algebra \\v is one of the simplest example of \nelliptic algebras \\cite{rFIJKMY,rFFr}, since we have elliptic theta-functions\nin the operator product expansion (OPE) formulas.\nWe strongly hope that this Virasoro-type algebra \nwill be constructed in a canonical way from \nthe elliptic algebra ${\\cal A}_{q,p}$ \\cite{rFIJKMY}\\footnote{\nThe relations between the parameters in ${\\cal A}_{q,p}$ and \\v \nare $q_{\\!\\!~_{{\\cal V}ir}}=p_{\\!\\!\\!~_{{\\cal A}}}$ and\n$p_{\\!\\!~_{{\\cal V}ir}}={q_{\\!\\!\\!~_{{\\cal A}}}}^2$.}\nthrough a Sugawara-type construction and \nthat will give us a clue for the total understanding of \\v.\n\nIn this review, we will explain how this two parameter \nVirasoro-type algebra \\v\narose in the CS-type model, and\nanother aim is to accumulate\nas many problems and applications of \\v as possible.\nThis paper is organized as follows.\nIn Section 2, we study basic results of the Virasoro-type algebras \nstarting from the definition of the two-parameter Virasoro-type algebra \\v. \nIn Section 3, a Heisenberg realization of the Virasoro-type algebra and \nits applications to the CS-type models are presented.\nBased on this realization, we discuss\nrepresentation theories, futher applications and \nrelations with other several models, in Section 4.\nSection 5 is devoted to summary and comments.\n\n\\section{Quantum deformed Virasoro algebra \\v }\nIn this section we examine some of the fundamental properties of \nthe Virasoro-type algebra\n\\v \\cite{rSKAO} \nwhich can be directly derived from the defining relation. \nA Heisenberg realization and its application to \nvarious problems are studied in the next sections.\n\\subsection{definition of \\v}\nLet $p$ be a generic complex parameter with $|p| < 1$.\nLet us consider an associative algebra generated by\n$\\{T_n|n\\in \\bf{Z}\\}$ with the relation\n$$\nf(w\/z)T(z)T(w)-T(w)T(z)f(z\/w)\n= {\\rm const.}\\left[\n \\delta \\Bigl(\\frac{pw}{z}\\Bigr)-\n \\delta \\Bigl(\\frac{p^{-1}w}{z}\\Bigr)\\right],\n$$\nwhere $T(z)=\\sum_{n\\in\\bf{Z}}T_n z^{-n}$,\n$\\delta(x)=\\sum_{n \\in {\\bf Z}}x^n$ and \n$f(z)=\\pzs{l}f_l z^l$ is a structure function. We show that the \ncommutativity of the diagram (Yang-Baxter equation for $T(z)$)\n\\begin{eqnarray}\n\\begin{array}{ccc}\n T(x)T(y)T(z) &{\\displaystyle \\mathop{\\llra}^{f(z\/y)\\times} } &\nT(x)T(z)T(y) f(y\/z) \\\\\n& & + T(x)\\; \\delta{\\rm-function}\\\\\n{\\scriptstyle f(y\/x)\\times} \\Biggl\\downarrow & & \n{\\scriptstyle f(z\/x)\\times} \\Biggl\\downarrow \\\\\n& & \\\\\nT(y)T(x)T(z) f(x\/y) & & \nT(z)T(x)T(y) f(y\/z) f(x\/z)\\\\\n+ T(z)\\; \\delta{\\rm-function} & & + T(x)\\; \\delta{\\rm-function}\\\\\n& & + T(y)\\; \\delta{\\rm-function}\\\\\n{\\scriptstyle f(z\/x)\\times} \\Biggl\\downarrow & & \n{\\scriptstyle f(y\/x)\\times} \\Biggl\\downarrow \\\\\n& & \\\\\nT(y)T(z)T(x) f(x\/y) f(x\/z) &{\\displaystyle \\mathop{\\llra}^{f(z\/y)\\times} } & \nT(z)T(y)T(x) f(x\/y) f(x\/z)f(y\/z)\\\\\n+ T(z)\\; \\delta{\\rm-function} & & + T(z)\\; \\delta{\\rm-function}\\\\\n+ T(y)\\; \\delta{\\rm-function} & & + T(y)\\; \\delta{\\rm-function} \\\\\n& & + T(x)\\; \\delta{\\rm-function} \n\\end{array}\\label{YB-for-T}\n\\end{eqnarray}\ndetermines this \nstructure function $f(z)$ completely. \nHere the terms denoted by ``$\\delta$-function'' \nmean some combinations of \nthe $\\delta$-functions and the structure functions.\nThe commutativity of this diagram means\n\\begin{eqnarray}\n0&=&T(x) \\Biggl( \\delta(pz\/y) g(x\/z)-\\delta(py\/z) g(x\/y) \\Biggr) \\nonumber\\\\\n &+&T(y) \\Biggl( \\delta(px\/z) g(y\/x)-\\delta(pz\/x) g(y\/z) \\Biggr) \\label{asso}\\\\\n &+&T(z) \\Biggl( \\delta(py\/x) g(z\/y)-\\delta(px\/y) g(z\/x) \\Biggr), \\nonumber\n\\end{eqnarray}\nwhere\n$g(x) \\equiv f(x)f(x\/p) - f(1\/x) f(p\/x)$.\nNote that $g(x)=-g(p\/x)$.\nIt is an interesting exercise to see that the general solution to the \neq.\\ (\\ref{asso}) is\n$g(x)= c_1 \\Bigl(\\delta(x\/p)-\\delta(x) \\Bigr)$,\nwhere $c_1$ is a constant. \nHereafter, we just set $c_1=1$,\nsince there is no loss of generality.\nNote that the Yang-Baxter equation for $T(z)$ is ``not'' \ntrivially satisfied even if the current $T(z)$ has\nno spin degrees of freedom,\nbecause we have the $\\delta$-function term in the \nrelation (\\ref{e:a1.2}).\n{}From this we have\n\\begin{eqnarray*}\nf(x)f(x p)= \\alpha + \\sum_{n=1}^\\infty (1-p^{n})x^n ,\n\\end{eqnarray*}\nwhere $\\alpha$ is a constant.\nIt should be noted that this is the place where one more parameter comes in.\nIf we parameterize $\\alpha$ by introducing another parameter $q$ as\n$$\n\\alpha={1-p \\over (1-q)(1-t^{-1})}, \\qquad\\quad t=qp^{-1},\n$$\nwe have the difference equation \n$$\nf(x)f(x p)=\\alpha { (1-qx)(1-t^{-1} x) \\over (1-x)(1-p x)}.\n$$\nThis can be solved as\n\\begin{equation}\nf(x)=\\exp \\Biggl\\{ \\sum_{n=1}^\\infty \n{ (1-q^n)(1-t^{-n}) \\over 1+p^n} { x^n \\over n} \\Biggr\\}. \\label{structure}\n\\end{equation}\nWe arrive at the definition of the quantum deformed Virasoro algebra\n\\v \\cite{rSKAO}.\n\n\\proclaim Definition 1.\nLet $p$ and $q$ be complex parameters with\nthe conditions $|p|<1$ and $|q|<1$, and set $t=qp^{-1}$.\nThe associative algebra \\v is generated by the current\n$T(z)=\\sum_{n \\in {\\bf Z}} T_n z^{-n}$ satisfying the relation\n\\begin{equation}\nf(w\/z)T(z)T(w)-T(w)T(z)f(z\/w)\n= -\\frac{(1-q)(1-t^{-1})}{1-p} \\left[\n \\delta \\Bigl(\\frac{pw}{z}\\Bigr)-\n \\delta \\Bigl(\\frac{w}{pz}\\Bigr)\\right],\n\\label{e:a1.2}\n\\end{equation}\nwith the structure function (\\ref{structure}).\n\n\n\\noindent Note that the constant factor in the R.H.S. is so chosen \nthat our Heisenberg realization of this algebra becomes simple.\n\nWe now have the associative algebra \\v \nwhich exists in a nontrivial way,\nsince we have so chosen the structure function $f(z)$\nthat \nthe Yang-Baxter equation for the \\v current\n(\\ref{YB-for-T}) will not give us any more \nrelations than the quadratic relation (\\ref{e:a1.2}) for $T(z)$.\n\nThe defining relation (\\ref{e:a1.2}) can be written in terms of $T_n$ as \n\\begin{equation}\n[T_n \\, , \\, T_m]=-\\ps{l}f_l\\left(T_{n-l}T_{m+l}-T_{m-l}T_{n+l}\\right)\n-\\frac{(1-q)(1-t^{-1})}{1-p}(p^{n}-p^{-n})\\delta_{m+n,0},\n\\label{e:a1}\n\\end{equation}\nwhere $[A,B]=AB-BA$.\n\nThe relation \\eq{e:a1.2} is invariant\nunder the transformations\n\\begin{eqnarray}\n{\\rm (I)}&& \\qquad\\qquad T_n \\to -T_n, \n\\label{e:a1.3} \\\\\n{\\rm (II)}&& \\qquad\\qquad (q,t)\\to (q^{-1},t^{-1}),\n\\label{e:aa1} \\\\\n{\\rm (III)} &&\\qquad\\qquad q \\leftrightarrow t. \\label{e:aa2}\n\\end{eqnarray}\n\\par\n\nIn what follows, \nwe will frequently use the notation\n\\begin{equation}\nt=qp^{-1}=q^\\beta.\n\\end{equation}\nThis parameter $\\beta$ plays the role of the coupling constant of\nthe Calogero-Sutherland model. See Section \\ref{secCS}.\n\n\n\n\\subsection{special limits of \\v}\nHere we study some special limits of \\v, which \nexplains the connections among known examples of the \nVirasoro-type algebras.\n\n\n\\subsubsection{limit of $q\\rightarrow 1$: ordinary Virasoro algebra}\nLet us study the limit $q\\to 1$ ($\\beta$: fixed) by\nparameterizing $q=e^{h}$. \nSuppose that $T(z)$ has the following expansion in $h$\n\\begin{equation}\nT(z)=2+\\beta \\left( z^2 L(z)+\\frac{(1-\\beta)^2}{4\\beta} \\right)h^2\n + T^{(2)}(z)h^4 +\\cdots.\n\\label{pe:a7}\n\\end{equation}\nThis expansion is consistent with the invariance under transformation \n\\eq{e:aa1}.\nThe defining relation \\eq{e:a1.2} gives us the well known relations for\nthe ordinary Virasoro current\n$ L(z)=\\sum_{n \\in {\\bf Z}}L_{n}z^{-n-2}$, namely\n\\begin{equation}\n[L_n , L_m]=(n-m)L_{n+m}+\\frac{c}{12}(n^3-n)\\delta_{n+m,0},\n\\label{e:a8}\n\\end{equation}\nwhere the central charge $c$ is\n\\begin{equation}\nc=1-\\frac{6(1-\\beta)^2}{\\beta}.\n\\label{e:a9}\n\\end{equation}\nThis relation between the central charge of\nthe Virasoro algebra and the coupling constant\nof CS model is discussed in Section \\ref{secCS}.\n\n\\subsubsection{Frenkel-Reshetikhin's $q$-Virasoro algebra\n($\\beta\\rightarrow 0$)}\nLet us consider the limit $\\beta\\to 0$ ($q$: fixed).\nIn this limit, we obtain the classical $q$-Virasoro algebra\nfound by Frenkel and Reshetikhin \\cite{rFR}. Their algebra is \nthe first one among many $q$-deformed Virasoro algebras,\nwhich was obtained through the study of the quantum affine algebra\n$U_q(\\widehat{sl}_2)$ \\cite{rAOS2}.\nOther nice features of their algebra are\nthat it is no more a Lie algebra but a quadratic algebra \nwhich resembles the relation\n$\\{L(u),L(v)\\}=[r(u-v),L(u)L(v)]$ \nin the quantum inverse scattering method,\nand there exists a mysterious \nresemblance between their bosonic realization and the \nBaxter's\ndressed vacuum form in the Bethe ansatz method. \nSee Section \\ref{secSYMMETRIC}.\n\nIn this limit, $T_n$'s become commutative. \nHowever, we can define the Poisson bracket structure\ndefined by $\\{\\;,\\;\\}_{\\rm P.B.}=-\\lim_{\\beta\\rightarrow 0}\n[\\;,\\;]\/(\\beta \\ln q )$.\nThus we have\n\\begin{eqnarray}\n\\{T_n,T_m\\}_{\\rm P.B.}=\n\\sum_{l\\in {\\bf Z}}{1-q^l \\over 1+q^l} T_{n-l}T_{m+l}\n+ (q^n-q^{-n}) \\delta_{n+m,0}, \\label{qvirFR}\n\\end{eqnarray}\nwhich is the relations for the classical $q$-Virasoro algebra \\cite{rFR}.\n\nIn the paper \\cite{rFr}, deformations of the KdV hierarchy\nis studied. The $N$-th Korteweg-de Vries (KdV) hierarchy\nis a bihamiltonian integrable system with the \n$N$-th order differential operators.\nThey are deformed to the $q$-shift operators\n$$\n D^N- t_1(z) D^{N-1}+\\cdots +(-1)^{N-1} t_{N-2}(z) D\n+(-1)^N t_{N-1}(z),\n$$\nwhere $D\\cdot f(z)=f(qz)$.\nIt was shown there that the deformed bihamiltonian structure\nis given by the Poisson bracket for the \n$q$-${\\cal W}$ algebra of Frenkel and Reshetikhin \\cite{rFR}.\n\\proclaim Proposition 1. \\hspace{-2mm}\\cite{rFR}\nIn the case of $N=2$, the bihamiltonian structure is \ngiven by\n\\begin{eqnarray}\n\\{t_i(z),t_j(w)\\}_1&=&\\delta\\left( {wq\\over z}\\right)-\n\\delta\\left( {w\\over zq}\\right),\\\\\n\\{t_i(z),t_j(w)\\}_2&=&\\sum_{m \\in{\\bf Z}} \\left( {w\\over z}\\right)^m\n{1-q^m \\over 1+q^m} t(z)t(w)+\n\\delta\\left( {wq\\over z}\\right)-\n\\delta\\left( {w\\over zq}\\right).\n\\end{eqnarray}\n\n\\noindent\nThese Poisson brackets coincide with that of the \n$q$-Virasoro algebra of Frenkel and Reshetikhin (\\ref{qvirFR})\nwith two different choices of the structure function $f(z)$:\n{\\it i.e.}, $f(z)=1$ and $f(z)$ given by (\\ref{structure})\nwith $\\beta\\rightarrow 0$.\n\n\\subsubsection{Zamolodchikov-Faddeev algebra of sine-Gordon and XYZ models}\nThe $q$-Virasoro algebra can be interpreted as \nthe Zamolodchikov-Faddeev (ZF) algebra satisfied by a \nparticle excitation operator.\nThe structure function $f(x)$ determined by the associativity\nrelates with \nthe $S$ matrix characterized by the factorization property.\n\nFirst, we can rewrite the defining relation of the \n$q$-Virasoro algebra (\\ref{e:a1.2})\nas follows;\n\\begin{equation}\\label{eq:ZFXYZ}\nT(z_1) T(z_2) \n=\n S\\left( {z_1\\over z_2}\\right) T(z_2) T(z_1)\n+ C\\left( \\delta\\left( {z_1\\over pz_2}\\right) \n+ \\delta\\left( {pz_1\\over z_2}\\right) \\right), \n\\end{equation}\nwith\n\\begin{eqnarray}\nS(z) \n&\\!\\!\\!=\\!\\!\\!&\n{f(z) \\over f(z^{-1})}=\n {\\vartheta_1(zt^{-1};p) \\,\\vartheta_0(zt ;p) \\over \n \\vartheta_1(zt ;p) \\,\\vartheta_0(zt^{-1};p) },\\\\\nC \n&\\!\\!\\!=\\!\\!\\!&\n{(1-q)(1-t^{-1})\\over (1-p)f(p)}=\n {( q;p^2)_\\infty (t^{-1};p^2)_\\infty \\over \n (pq;p^2)_\\infty (pt^{-1};p^2)_\\infty},\n\\end{eqnarray}\nfor $|p|<1$.\nHere, \n$\\vartheta_1(z;p) = \n-ip^{1\\over 4} z^{1\\over 2}\n(p^2 z;p^2)_\\infty$$( z^{-1};p^2)_\\infty$$(p^2;p^2)_\\infty$ and\n$\\vartheta_0(z;p) = \n(p z;p^2)_\\infty \\times$ $\\times(pz^{-1};p^2)_\\infty$$(p^2;p^2)_\\infty$\nwith $(z;q)_\\infty \\equiv \\prod_{n\\geq 0}(1-zq^n)$.\n\nNext, let $p = e^{\\tau\\pi i}$\nand perform a modular transformation \n$\\tau\\rightarrow -1\/\\tau$ for theta functions\nand take a limit $-1\/\\tau \\rightarrow i\\infty$, {\\it i.e.}\n$p\\rightarrow 1$. Changing the parameterization as \n$z=p^{i{\\theta\\over\\pi}}$ and $t=p^{\\xi}$, we have \n\n\\proclaim Proposition 2. \n\\hspace{-2mm}\\cite{rL}\\footnote{The notations in \\cite{rL} are\n$x=p^{1\\over 2}$, $\\xi=\\beta\/(1-\\beta)$ and $\\epsilon=i\\tau$.}~\nIn the limit of $p\\rightarrow 1$, \nthe $q$-Virasoro generator satisfies \nthe following Zamolodchikov-Faddeev algebra\\footnote{\nThis ZF-equation should be understood in the sense of analytic continuation.}\n of the sine-Gordon model,\n\\begin{equation}\\label{eZFSG}\n{\\cal T}(\\theta_1) {\\cal T}(\\theta_2) = S\\left( {\\theta_1- \\theta_2}\\right) \n{\\cal T}(\\theta_2) {\\cal T}(\\theta_1), \\qquad\nS(\\theta) = {\\sinh\\theta + i\\sin\\pi\\xi \\over \\sinh\\theta - i\\sin\\pi\\xi},\n\\end{equation}\nwhere ${\\cal T}(\\theta)=\\lim_{-1\/\\tau \\rightarrow i\\infty}T(z)$.\n\n\\noindent\nNamely,\nthe $p=1$ $q$-Virasoro generator ${\\cal T}(z)$ creates \nthe basic particle (the first breather) with a rapidity $\\theta$\nof the sine-Gordon model, defined by the following Lagrangian density\n\\begin{equation}\\label{eLagrangianSG}\n{\\cal L}_{SG} = \n{1\\over2}(\\partial_\\mu\\phi)^2 + \n\\left( {m_0\\over b}\\right)^2 \\cos\\left( b\\phi\\right),\\qquad \nb^2=8\\pi\\beta=8\\pi{\\xi\\over 1+\\xi},\n\\end{equation}\nat the attractive range $0<\\xi<1$.\nThis identification is based on the \ncoincidence of the $S$-matrix for the \nbasic scalar particles in sine-Gordon model with\n$f(z)\/f(1\/z)$ in the limit of $p\\rightarrow 1$, and \nthe existence of\nsimple poles at the points $\\theta_1 = \\theta_2 \\pm i\\pi$\nin the delta function terms.\nFor the axiom of the ZF operator, see \\cite{rL2}.\nLukyanov also proposed in \\cite{rL} that,\nfor generic $p$,\neq.\\ \\eq{eq:ZFXYZ} can be interpreted as the ZF relation \nof the XYZ model,\n{\\it i.e.},\nthe $q$-Virasoro generator is \nthe basic scalar creation operator of this model.\n\n\\subsubsection{limit of $q\\rightarrow 0$}\nNext, study the $q\\rightarrow 0$ limit ($t$: fixed).\nThis limit is interesting from the point of view of the \nrepresentation theory of the Hall-Littlewood polynomials \\cite{rM}\nin terms of the Heisenberg algebra.\nIt is known by the work of Jing \\cite{rJ}\nthat the Hall-Littlewood polynomials are \nrealized by a multiple integral formula.\nThe integration kernel is \nsimply given by multiplying vertex \noperators on a vacuum state.\nAs for the number of the integration variables\nand the vertex operators, it is related with the \nshape of the Young diagram of each Hall-Littlewood polynomial.\nHis realization can be regarded as a deformation of \nthe determinant representation of the Schur polynomials.\nRecently, similar integral \nrepresentations are studied\nfor the Jack polynomials and the \nMacdonald polynomials \\cite{rSt,rM,rMY,rAMOS,rAOS}. \nHowever, the number of the integrals \nthere is much greater in general \nthan the case of the Schur or Hall-Littlewood\npolynomials. We will discuss this in \nSection \\ref{q0limit}\nby using the Heisenberg realization of \\v in the limit $q\\rightarrow0$.\n\nSo as to obtain well behaving generators at $q \\to 0$, \nlet us scale $T_n$ as \n\\begin{equation}\n{\\tilde T}_n = T_n p^{\\frac{|n|}{2}}.\n \\label{r1}\n\\end{equation}\nUsing this notation and \ntaking the limit ($q \\to 0$) of the relation (\\ref{e:a1.2}),\nwe have the commutation relation for \nthe deformed Virasoro algebra in this limit.\n\\proclaim Proposition 3. The commutation relations for \nthe deformed Virasoro generators\n${\\tilde T }_n$ are \n\\begin{eqnarray}\n\\left[{\\tilde T }_n,{\\tilde T }_m\\right] &=&\n-(1-t^{-1})\\sum_{\\ell =1}^{n-m}{\\tilde T }_{n-\\ell}{\\tilde T }_{m+\\ell}\n\\quad \\mbox{for}\n\\quad n > m > 0 \\quad \\mbox{or}\\quad 0>n>m, \n\\nonumber\\\\ \n\\left[{\\tilde T }_n,{\\tilde T }_0\\right] &=&\n -(1-t^{-1})\\sum_{\\ell =1}^{n}{\\tilde T }_{n-\\ell}{\\tilde T }_{\\ell}\n-(t-t^{-1})\\sum_{\\ell =1}^{\\infty} t^{-\\ell}\n{\\tilde T }_{-\\ell}{\\tilde T }_{n+\\ell} \\quad \\mbox{for}\\quad n > 0, \n\\nonumber\\\\ \n\\left[{\\tilde T }_0,{\\tilde T }_m\\right] &=&\n -(1-t^{-1})\\sum_{\\ell =1}^{-m}{\\tilde T }_{-\\ell}{\\tilde T }_{m+\\ell}\n-(t-t^{-1})\\sum_{\\ell =1}^{\\infty} t^{-\\ell}\n{\\tilde T }_{m-\\ell}{\\tilde T }_{\\ell} \\quad \\mbox{for}\\quad 0 > m, \n\\nonumber\\\\ \n\\left[{\\tilde T }_n,{\\tilde T }_m\\right] &\\!\\!=\\!\\!&\n-(1-t^{-1}){\\tilde T }_{m}{\\tilde T }_{n} \n-(t-t^{-1})\\sum_{\\ell =1}^{\\infty} t^{-\\ell}\n{\\tilde T }_{m-\\ell}{\\tilde T }_{n+\\ell} \n\\nonumber\\\\ \n& & +(1-t^{-1})\\mbox{\\rm sign}(n)\\delta_{n+m,0}\n\\quad \\mbox{for}\\quad n> 0> m, \n\\label{r2}\n\\end{eqnarray}\nwhere the function $\\mbox{\\rm sign}(n)$ is\n$1$, $0$ and $-1$ for $n>0$, $n=0$ and $n<0$, respectively.\n\n\n\\noindent\n\n\n\\subsection{highest weight modules of \\v}\nLet us define the Verma module of \\v.\nLet $\\kv{\\lambda}$ be the highest weight vector\nsuch that\n$T_0 \\kv{\\lambda}=\\lambda\\kv{\\lambda}$, $\\lambda\\in{\\bf C}$ and \n$T_n \\kv{\\lambda}=0$ for $n>0$.\nThe Verma module $M(\\lambda)$ is defined by\n$M(\\lambda)=\\mbox{\\v}\\kv{\\lambda}$.\nThe irreducible highest module $V(\\lambda)$ is obtained from\n$M(\\lambda)$ by removing all singular vectors and their descendants.\nRight modules are defined in a similar way from the\nlowest weight vector\n$\\bv{\\lambda}$\ns.t.\n$\\bv{\\lambda}T_0=\\lambda\\bv{\\lambda}$ and\n$\\bv{\\lambda}T_n=0$ for $n<0$.\nA unique invariant paring is defined by setting\n$\\bv{\\lambda}\\lambda \\rangle = 1$.\nThe Verma module $M(\\lambda)$ may have\nsingular vectors same as that of the ordinary Virasoro algebra.\nLet us introduce the (outer) grading operator $d$ which satisfies\n$[d , T_n]= n T_n$ and set $d\\ket{\\lambda}=0$. We call a vector\n$\\ket{v} \\in M(\\lambda)$ of level $n$ if $d\\ket{v}=-n\\ket{v}$.\n\nWhether there exist the singular vectors or not is\nchecked by calculating the Kac determinant.\nHere, we give some explicit forms of $f_{n}$\nwhich we will use for the calculations\n\\begin{eqnarray}\n& & f_{1}=\\frac{(1-q)(1-t^{-1})}{1+p},\\nonumber\n\\label{e:a11.2}\n\\\\\n& &\nf_{2}=\\frac{(1-q^{2})(1-t^{-2})}{2(1+p^{2})}+\n\\frac{(1-q)^2(1-t^{-1})^2}{2(1+p)^2}. \\nonumber\n\\label{e:a11.3}\n\\end{eqnarray}\n\\par\nAt level 1, the Kac determinant is the $1\\times 1$\nmatrix as follows\n\\begin{equation}\n\\bv{\\lambda}T_{1} T_{-1}\\kv{\\lambda}\n=\\frac{(1-q)(1-t)}{q + t}(\\lambda^2 - (p^{1\/2}+p^{-1\/2})^2 ).\n\\label{e:a15}\n\\end{equation}\nTherefore, there exist a singular vector at level 1 iff\n$\\lambda=\\pm \\left(p^{1\/2}+p^{-1\/2}\\right)$,\n since $q$ and $t$ are generic.\nThe signs $\\pm$ in the RHS are due to the symmetry \\eq{e:a1.3}.\n\\par\nAt level 2, the Kac determinant is\n\\begin{eqnarray}\n\\& \\hskip-10truem\n \\left|\n \\begin{array}{clcr}\n \\bv{\\lambda}T_{1}T_{1}T_{-1}T_{-1}\\kv{\\lambda} &\n \\bv{\\lambda}T_{1}T_{1}T_{-2}\\kv{\\lambda} \\\\\n \\bv{\\lambda}T_{2}T_{-1}T_{-1}\\kv{\\lambda} &\n \\bv{\\lambda}T_{2}T_{-2}\\kv{\\lambda} \\\\\n \\end{array}\n \\right|\n\n \\frac{(1-q^2)(1-q)^2 q^{-4}(1-t^2)(1-t)^2 t ^{-4}}{(q +t)^2 (q^2 +t^2)}\n\\nonumber \\\\\n\\&\\hskip35truemm\n\\times (\\lambda^2 qt-(q+t)^2 )( \\lambda^2 q^2t-(q^2+t)^2 )\n ( \\lambda^2 qt^2 - (q+t^2)^2 ).\n\\label{e:a18}\n\\end{eqnarray}\nThe vanishing conditions of the Kac determinant are\n\\begin{eqnarray}\n& \\mbox{(i)}&\\lambda =\\pm \\left(p^{1\/2} + p^{-1\/2}\\right),\n\\label{e:a19.1} \\\\\n& \\mbox{(ii)}&\\lambda =\\pm \\left(p^{1\/2}q^{1\/2} + p^{-1\/2}q^{-1\/2}\\right),\n\\label{e:a19.2} \\\\\n& \\mbox{(iii)}& \\lambda =\\pm \\left(p^{1\/2}t^{-1\/2} + p^{-1\/2}t^{1\/2}\\right).\n\\label{e:a19.3}\n\\end{eqnarray}\nIn the case (i), there is a singular vector at level 1.\nIn the cases (ii) and (iii), we have a singular vector at level 2.\nThe singular vector for the case (ii) is\n\\begin{equation}\n \\frac{qt^{-1\/2}(q+t)}{(1-q)^2(1+q)}T_{-1}T_{-1}\\kv{\\lambda}\n\\mp T_{-2}\\kv{\\lambda},\n\\label{e:a20.1}\n\\end{equation}\nand for (iii) is\n\\begin{equation}\n \\frac{q^{-1\/2}t(q+t)}{(1-t)^2(1+t)}T_{-1}T_{-1}\\kv{\\lambda}\n\\mp T_{-2}\\kv{\\lambda}.\n\\label{e:a21.1}\n\\end{equation}\n\n\nTo calculate the Kac determinant becomes difficult task\nwhen $N$ increases.\nWe have calculated up to level 4, and write down the\nconjectural form at level $N$.\n\n\\proclaim Conjecture 1. The Kac determinant at level-N is written as \n\\begin{equation}\\label{Kacconj}\n \\det{}_N\n =\n \\det\\Bigl(\\langle i\\ket{j}\\Bigr)_{1\\leq i,j\\leq p(N)}\n \\!\\!=\\!\\!\n \\prod_{\\scriptstyle r,s\\geq 1 \\atop \\scriptstyle rs\\leq N}\n \\Bigl(\\lambda^2-\\lambda_{r,s}^2\\Bigr)^{p(N-rs)}\n \\left(\\frac{(1-q^r)(1-t^r)}{q^r+t^r}\n \\right)^{p(N-rs)},\n\\end{equation}\nwhere \n$\n\\lambda_{r,s}=\nt^{r\/2}q^{-s\/2}+t^{-r\/2}q^{+s\/2}\n$\nand\nthe basis at level $N$ is defined\n$\\ket{1}=T_{-N}\\ket{\\lambda}$,\n$\\ket{2}=T_{-N+1}T_{-1}\\ket{\\lambda}$,$\\cdots,\n\\ket{p(N)}=T_{-1}^N\\ket{\\lambda}$, and $p(N)$ is the\nnumber of the partition of $N$.\n\n\\noindent\nWe remark that the $\\lambda$ dependence has essentially\nthe same structure as the case of the usual Virasoro algebra.\nTherefore, if $q$ and $t$ are generic, the character of the quantum Virasoro\nalgebra \\v,\nwhich counts the degeneracy at each level,\nexactly coincides with that of the usual Virasoro algebra.\nThe $\\lambda$-independent factor\nin the RHS will play an important role when we study the case\nthat $q$ is a root of unity.\n\n\\subsection{problem of obtaining a geometric interpretation of \\v}\nIt is remarkable that \\v arises in a variety of off-critical models\nin a universal way.\nAs for the geometric interpretation, however, \nwe have not\nhave a satisfactory answer yet.\nFor the ordinary Virasoro algebra with $c=0$, we have \nthe differential operator realization,\n$L_n=-z^{n+1} \\partial_z$,\nwhich explains that \nthe Virasoro algebra describes the Lie algebra structure \nof the tangent space of the conformal group.\nAs a natural deformation of this differential operator realization,\nis it possible to have a difference operator representation of\nthe deformed Virasoro algebra \n\\v for some parameters $q$ and $p$?\nIt may be possible to study a connection between \\v\nand the analysis over the local fields in the limit of \n$q\\rightarrow 0$ with fixed $t$. See Section \\ref{q0limit}.\n\n\n\\section{Free boson realization of \\v }\nIn this section, we present the Heisenberg realization of \\v\nand its applications to Calogero-Sutherland-type models.\n\n\n\\subsection{conformal field theory}\nOne of the simplest example of the conformal field theory is \nthe massless Klein-Gordon field in $1+1$ dimensions. \nWe briefly review how we can treat the \nVirasoro current in terms of the Klein-Gordon \nfield at the conformal point to \nprepare basic ideas and notations for the later discussions.\nAs for the detail, the readers are referred to the original or\nreview articles of the conformal field theory \\cite{rBPZ,rG}.\nThe \naction for the Klein-Gordon field $\\phi(x,\\tau)$ is \n$$\nS_{Eucl}=\\int d\\tau dx{1 \\over 2} \\left(\n (\\partial_\\tau \\phi)^2+ (\\partial_x \\phi)^2\n+ m^2 \\phi^2\\right).\n$$\nIf the system is massless $m=0$ then it acquires the infinite dimensional \nconformal symmetry. \nWe shall see how the generators of this conformal \ntransformation are realized in terms of the \nKlein-Gordon field $\\phi(x,\\tau)$.\nLooking at the \nequation of motion\n\\begin{eqnarray}\n\\partial_w \\partial_{\\bar w}\\phi(w,\\bar{w})=0,\\qquad\\qquad (w=x+i \\tau),\n\\end{eqnarray} \nwe have the decoupling of $\\phi$ into \nchiral and anti-chiral parts as\n$$\n\\phi(w,\\bar{w})=a(w)+\\bar{a}(\\bar{w}).\n$$\nTherefore,\nwe can study the chiral part and anti-chiral part separately.\nAfter the compactification of the \nspace into the segment \n$0\\leq x \\leq 2\\pi$ with the periodic boundary condition \nand introducing the conformal mapping $z=e^{iw}$, we arrive at\nthe expansion\n\\begin{eqnarray}\na(z)=Q+a_0\\ln z - \\sum_{n\\neq 0} {a_n \\over n} z^{-n}.\n\\end{eqnarray}\nThe Poisson brackets for these modes are\n\\begin{eqnarray}\n&& \\{ a_n,a_m\\}_{\\rm P.B.}=n \\delta_{n+m,0}\\qquad\n\\{ a_n,Q\\}_{\\rm P.B.}= \\delta_{n,0}.\n\\end{eqnarray}\nThe Virasoro current $L(z)$ is written as\n$$\nL(z)= \\sum_{n\\in{\\bf Z}} L_n z^{-n-2}=\n{1 \\over 2} \\left( \\partial a(z) \\right)^2.\n$$\nUsing the formula\n$$\n\\{ \\partial a(z),\\partial a(w)\\}_{\\rm P.B.}= {1 \\over z^2}\\delta'(w\/z),\n$$\nwe obtain\n\\begin{equation}\n\\{ L_n,L_m\\}_{\\rm P.B.}= (n-m) L_{n+m}.\n\\end{equation}\n\n\nTo quantize the system, we replace the Poisson brackets by\nthe commutators as\n\\begin{equation}\n[a_n,a_m]= n \\delta_{n+m}\\qquad [ a_n,Q]= \\delta_{n,0},\n\\end{equation}\nand define the Virasoro current with the\nnormal ordered product as\n$$\nL(z)= {1 \\over 2} :\\left( \\partial a(z) \\right)^2:.\n$$\nThe definition of this normal ordering is that \nwe shift every \nannihilation operators ({\\it i.e.}, $a_n$ with $n\\geq 0$)\nto the right of \ncreation operators ({\\it i.e.}, $a_n$ with $n<0$ and $Q$).\nFor example $:a_{-1}a_{2}:=a_{-1}a_{2}$,\n$:a_{3}a_{-2}:=a_{-2}a_{3}$ and so on.\nThis quantized Virasoro current obeys eq.\\ \\eq{e:a8} with $ c=1$.\nIn general the central charge $c$\ndepends on the model; $c=1\/2$ for real fermion, $c=3k\/(k+2)$ for \n$\\widehat{su}(2)_k$\nKac-Moody algebra, for example.\nOne more important construction of the \nVirasoro algebra which is relevant to our later discussion is\nthe Feigin-Fuchs construction\n$$\nL(z)= {1 \\over 2} :\\left( \\partial a(z) \\right)^2:+\n\\alpha_0 \\partial^2 a(z).\n$$\nFor this realization we have the central charge less than one\n$$\nc=1-12 \\alpha_0^2,\n$$\nif $\\alpha_0$ is real.\nFor the later discussion we will parameterize $\\alpha_0$ as\n$$\n\\alpha_0 =\\frac{1}{\\sqrt{2}}\\left({\\sqrt{\\beta}-\\sqrt{1\/\\beta}}\\right).\n$$\nWe will see that this parameter $\\beta $ has the meaning of the coupling\nconstant of the Calogero-Sutherland model. See Section \\ref{secCS}.\n\n\n\\subsection{singular vectors of the Virasoro algebra}\nWhat is very special in the representation space is\nthe singular vectors, which correspond to decoupled states from \nthe physical space. We review some of the \nexplicit formulas of the singular vectors.\n\nThe highest weight state $|h\\rangle$ is defined by\n$L_n |h\\rangle=0$ for $n>0$ and\n$L_0 |h\\rangle=h |h\\rangle$,\nand the Verma module $M(h)$ is spanned over the highest weight state\n$|h\\rangle$ \nas $M(h)=\\langle L_{-1},L_{-2},\\cdots\\rangle |h\\rangle$.\nThe singular vector $|\\chi\\rangle \\in M(h)$ at level $n$ is defined by\n$L_n |\\chi\\rangle = 0$ for $n>0$ and \n$L_0 |\\chi\\rangle = (h+n)|\\chi\\rangle$.\nBy a standard argument, it has null norm with any states\nin the Verma module;\n$\\langle * | \\chi \\rangle =0$.\nThe existence of such state depends crucially on the\nchoice of parameter $c$ and $h$.\nCelebrated Kac formula shows that if they are\nexplicitly parameterized as eq.\\ \\eq{e:a9} and \n\\begin{equation}\\label{eq:Kac}\nh_{rs}=\\frac{(\\beta r-s)^2-(\\beta-1)^2}{4\\beta},\n\\label{eq:KacFormula}\n\\end{equation}\nfor an arbitrary parameter $\\beta(\\neq0)\\in{\\bf C}$\nand integers $r$ and $s$ with $rs>0$, \nthere exists unique (up to normalizatin) null state of level $rs$.\nSome of the lower lying states can be explicitly\nobtained by solving the defining conditions.\nLet $|\\chi_{rs}\\rangle\\in M(h_{rs})$ be \nthe null state at level $rs$.\nFor example,\nwe obtain,\n\\begin{eqnarray}\\label{eq:Lower}\n|\\chi_{11}\\rangle&=& L_{-1}|h_{11}\\rangle,\\nonumber\\\\\n|\\chi_{12}\\rangle&=& (L_{-2}-{\\beta}L_{-1}^2)|h_{12}\\rangle,\\\\\n|\\chi_{22}\\rangle &=& \\left(L_{-4}+\n\\frac{2(\\beta^2-3\\beta+1)}{3(\\beta-1)^2}L_{-3}L_{-1}-\n\\frac{(\\beta+1)^2}{3\\beta}L_{-2}^2\\right.\\nonumber\\\\\n&&\\left.+\n\\frac{2(\\beta^2+1)}{3(\\beta-1)^2}L_{-2}L_{-1}^2-\n\\frac{\\beta}{3(\\beta-1)^2}L_{-1}^4\\right)\n|h_{22}\\rangle, \\nonumber \n\\end{eqnarray}\nand so on.\n\nSince we have the Feigin-Fuchs realization of $L(z)$,\nthe singular vectors are also written in\nthe bosonic creation oscillators $a_{-n}$ acting on the \nvacuum state $|\\alpha \\rangle$ \n({\\it i.e.}, $a_n|\\alpha\\rangle=0$ for $n> 0$ \nand $a_0|\\alpha\\rangle=\\alpha|\\alpha\\rangle$).\nIt is quite remarkable that all these singular vectors\ncan be regarded as the ``Jack symmetric polynomials''\nwhen we replace $a_{-n}$ by the power sum $\\sum_{i=1}^N x_i^n$.\nAs for the proof of this correspondence \nthe reader is referred to \\cite{rMY}.\n\nIn the Feigin-Fuchs construction, the Virasoro generators are \n\\begin{equation}\\label{eq:coulomb}\nL_n=\\frac{1}{2}\\sum_{m\\in {\\bf Z}} :a_{n+m} a_{-m}: -\\alpha_0\n (n+1) a_n.\n\\end{equation}\nThe highest weight state $|h_{rs}\\rangle$ is realized as \n$\\ket{\\alpha_{rs}}\\equiv e^{\\alpha_{rs}Q}|0\\rangle$,\nwith\n\\begin{equation}\n\\alpha_{rs}=\\frac{1}{\\sqrt 2}\\left((1+r)\\sqrt \\beta\n-(1+s)\\sqrt{1\/\\beta}\\right).\n\\label{eq:alphaRS}\n\\end{equation}\nIn terms of this free boson oscillators,\nthe null states (\\ref{eq:Lower}) are written as,\n\\begin{eqnarray}\n|\\chi_{11}\\rangle & \n& a_{-1} |\\alpha_{11}\\rangle, \\nonumber\\\\\n|\\chi_{12}\\rangle & \n& \\left(a_{-2}\n + \\sqrt{2\\beta}a_{-1}^2\\right)|\\alpha_{12}\\rangle, \\\\\n|\\chi_{22}\\rangle & \n& \\left( a_{-4}\n +\\frac{4\\sqrt{2\\beta}}{1-\\beta}\n a_{-3}a_{-1}\n -2\\frac{1+\\beta+\\beta^2}{\\sqrt{2\\beta}(1-\\beta)}a_{-2}^2\n -4 a_{-2}a_{-1}^2\n -\\frac{2\\sqrt{2\\beta}}{1-\\beta} a_{-1}^4\n \\right)|\\alpha_{22}\\rangle.\\nonumber\n\\end{eqnarray}\n\nTo translate these expressions into the Jack symmetric functions,\none can apply the rule,\n\\begin{equation}\na_{-n}\\rightarrow \\sqrt{\\frac{\\beta}{2}}\\sum_{i=1}^N x_i^n,\n\\qquad\n\\ket{\\alpha_{rs}}\\rightarrow 1.\n\\end{equation}\nUsing this rule, we have the correspondence \\cite{rMY}\n\\begin{equation}\n|\\chi_{rs}\\rangle \\sim J_{\\{ s^r\\}}(x;\\beta),\n\\end{equation}\nwhere the R.H.S. is the Jack symmetric polynomial for the \nrectangular diagram ${\\{ s^r\\}}$.\n\nIt was shown in \\cite{rAMOS} that the Jack polynomials for\narbitrary Young diagrams are realized as \nthe singular vector of the $W_N$ algebra.\n\n\n\n\\subsection{Calogero-Sutherland Hamiltonian, the Jack \npolynomials and the Virasoro algebra}\n\\label{secCS}\nIn this section, we explain the ``Calogero-Sutherland-Virasoro'' \ncorrespondence. As for the details, the readers are referred to \n\\cite{rAMOS} and references therein. \n\nThe Jack symmetric polynomials arise in \nthe Calogero-Sutherland (CS) model \\cite{rCS} as the wave functions of \nthe excited states of this model.\nAfter a suitable coordinate transformation, \nthe Hamiltonian and momentum of this system become\n\\begin{equation}\n {\\cal H}=\n \\sum_{i=1}^N D_i^2\n +\\beta\\sum_{i0$)\n\\begin{eqnarray}\na_n=\\sqrt{1 \\over 2\\beta}a^{\\rm old}_n ,\n\\qquad a_{-n}=\\sqrt{2 \\over \\beta}a^{\\rm old}_{-n}, \\qquad \na_0=\\sqrt{1 \\over 2\\beta}a^{\\rm old}_0,\\qquad\nQ= \\sqrt{2 \\over \\beta}Q^{\\rm old}.\n\\end{eqnarray}\nCorrespondingly we change the notation for \n$\\alpha_{rs}$ as\n$$\n\\alpha_{rs}=\\frac{1}{ 2}\\left((1+r)\\beta\n-(1+s)\\right),\n$$\nand write \n$|h_{rs}\\rangle= \\ket{\\alpha_{rs}}\\equiv e^{\\alpha_{rs}Q}|0\\rangle$\nas before.\n\nOne may derive bosonized Hamiltonian \nand momentum $\\widehat{\\cal H}$ and $\\widehat{\\cal P}$\nwhich satisfy\n$\n{\\cal O} \\pi_N \\langle 0|\\exp(\\beta \\sum_n{ a_n \\over n}p_n)\n=\\pi_N \\langle 0|\\exp(\\beta \\sum_n {a_n \\over n}p_n)\\widehat{\\cal O},\n\\label{e12}\n$\nwhere ${\\cal O}={\\cal H},{\\cal P}$, and \n$\\pi_N$ denotes the projection to the $N$-particle space.\nThey are given by,\n\\begin{equation}\n \\widehat{\\cal H}= \\beta \\sum_{n>0}\na_{-n} \\,L_n + (\\beta-1+\\beta N-2 a_0)\\, \\widehat{\\cal P},\n\\qquad\\quad\n\\widehat{\\cal P}=\\beta\\sum_{n=1}^\\infty a_{-n}a_n.\n\\end{equation}\nHere $L_n$'s are the annihilation operators of\nthe Feigin-Fuchs construction of the Virasoro algebra with\nthe center $c$ in \\eq{e:a9}.\nUsing these formulas, it is easily shown that\nthe singular vector $|\\chi_{rs}\\rangle$ of the Virasoro algebra is\nproportional to \nthe Jack polynomial for the Young diagram\n$\\lambda=\\{s^r\\}$ , since we have\n\\begin{equation}\n \\widehat{\\cal H}|\\chi_{rs}\\rangle =\n\\epsilon_{\\{s^r\\}}|\\chi_{rs}\\rangle.\n\\end{equation}\nwith $\\epsilon_{\\{s^r\\}}=(\\beta(N-r)+s)\\,rs$.\n\n\n\\subsection{screening currents}\nIn the Feigin-Fuchs construction we are able to have \ntwo weight-one primary fields $S_\\pm(z)$, which are called screening currents.\nThe condition of being weight-one primary gives us \nthe equation\n\\begin{eqnarray}\n[L_n,S_\\pm(z)]= \\partial_z \\left( z^{n+1} S_\\pm(z) \\right). \n\\end{eqnarray}\nWe have the solutions of this equation as follows\n\\begin{eqnarray}\n S_+(z)\n &\\!\\!=\\!\\!&\n \\exp\\left\\{\\ps{n}\\beta \\frac{a_{-n}}{n}z^{n}\\right\\}\n \\exp\\left\\{-\\ps{n}2\\beta \\frac{a_{n}}{n}z^{-n}\n \\right\\} e^{\\beta Q}z^{2\\beta a_0}, \\label{Vscr1}\\\\\n S_-(z)\n &\\!\\!=\\!\\!&\n \\exp\\left\\{-\\ps{n}\\frac{a_{-n}}{n}z^{n}\\right\\}\n \\exp\\left\\{\\ps{n}2\\frac{a_{n}}{n}z^{-n}\\right\\}\n e^{- Q}z^{-2 a_0}. \\label{Vscr2}\n\\end{eqnarray}\n\nIn the Fock module with the highest weight state\n$\\ket{\\alpha_{r,s}}$, we have a singular vector $\\ket{\\chi_{r,s}}$\nat level $rs$. By using a screening current $S_+(z)$,\n$\\ket{\\chi_{r,s}}$ is given as follows \\cite{rKM,rTK}:\n\\begin{eqnarray}\n \\ket{\\chi_{r,s}}\n &\\!\\!=\\!\\!&\n \\oint\\prod_{j=1}^r\\frac{dz_j}{2\\pi i}\\cdot\n \\prod_{i=1}^r S_+(z_i)\n \\ket{\\alpha_{-r,s}} \n\\label{eq:VirSing}\\\\\n &\\!\\!=\\!\\!&\n \\oint\\prod_{j=1}^r\\frac{dz_j}{2\\pi iz_j}\\cdot\n \\prod_{i,j=1 \\atop i0&& \\\\\n&&\\\\\nq\\mbox{-deformation} \\Biggl\\downarrow &&\nq\\mbox{-deformation} \\Biggl\\downarrow \\\\\n && \\\\\n {\\rm Macdonald\\;difference\\;operator} &\n{\\displaystyle\\mathop{ \\llla}^{\\rm singular\\;vectors}}& \nT_n\\;{\\rm generates\\; the\\;} q\\mbox{-Virasoro algebra \\v } \\\\\n\\widehat{D}_{q,t} = \\ps{n} \\psi_{-n}T_{n}+\\cdots &&\\\\\n T_n P_{\\{s^r\\}}(x;q,t)=0\\quad {\\rm for}\\;n>0&& \\\\\n &&\n\\end{array}\n\\end{eqnarray*}\nHere, $\\psi_{-n}$ should be a suitable combination of \nthe creation operators $a_{-m}$'s with degree $n$.\n\nThe problem is ``to make this diagram commutative.''\nHowever, we have many unknown operators \n$\\psi_{-n}$ and $T_n$. \nTo have enough data to solve this problem, \nsome knowledge of the $q$-deformed screening operators\nmust be needed.\n\nBy studying the action of the bosonized Macdonald operator\n$\\widehat{D}_{q,t}$,\nwe are able to obtain bosonized realization for some of \nthe Macdonald polynomials.\nLet,\n\\begin{equation}\n \\exp\\left\\{ \\sum_{n=1}^\\infty \\frac{1-q^{\\gamma n}}{1-q^n}\n \\frac{a_{-n} }{n} z^n\\right\\}\n =\n \\sum_{n=0}^{\\infty} \\widehat Q_n^{(\\gamma)} z^n \\label{macQ}\n\\end{equation}\nthe states $\\widehat Q_n^{(\\gamma)}|0\\rangle$ with $\\gamma=\\beta$ or $-1$\nare the Macdonald polynomials $Q_\\lambda(x;q,t)$\ncorresponding to the Young diagram\nwith single row $(n)$ or single column $(1^n)$, respectively.\nAs for the difference between $P_\\lambda(x;q,t)$ and \n$Q_\\lambda(x;q,t)$, see \\cite{rM}.\nWe obtained other examples;\nfor the Young diagram with\ntwo rows $\\lambda=(\\lambda_1,\\lambda_2)$ or\ntwo columns ${}^t\\lambda=(\\lambda_1,\\lambda_2)$ we have\n\\begin{eqnarray}\n \\widehat Q_{(\\lambda_1,\\lambda_2)}^{({\\gamma})}|0\\rangle\n &\\!\\!=\\!\\!&\n \\sum_{\\ell=0}^{\\lambda_2}c^{({\\gamma})}(\\lambda_1-\\lambda_2,\\ell)\n \\widehat Q_{\\lambda_1+\\ell}^{({\\gamma})}\n \\widehat Q_{\\lambda_2-\\ell}^{({\\gamma})}\n |0\\rangle, \\nonumber\\\\\n c^{({\\gamma})}(\\lambda,\\ell)\n &\\!\\!=\\!\\!&\n \\frac{1-q^{\\frac{\\beta}{\\gamma}(\\lambda+2\\ell)}}\n {1-q^{\\frac{\\beta}{\\gamma}(\\lambda+\\ell)}}\n \\prod_{j=1}^{\\ell}\n \\frac{1-q^{\\frac{\\beta}{\\gamma}(\\lambda+j)}}\n {1-q^{\\frac{\\beta}{\\gamma}j}}\\cdot\n \\prod_{i=1}^{\\ell}\n \\frac{q^{\\gamma}-q^{\\frac{\\beta}{\\gamma}(i-1)}}\n {1-q^{\\gamma+\\frac{\\beta}{\\gamma}(\\lambda+i)}},\n\\end{eqnarray}\nwith $\\gamma=\\beta$ or $-1$, respectively.\n The Macdonald polynomials of single hook $(n,1^m)$ are,\n\\begin{equation}\n \\widehat Q_{(n,1^{m})}|0\\rangle =\n \\sum_{\\ell=0}^{m} \\frac{1-q^{n+\\ell}t^{m-\\ell}}{1-q} q^{m-\\ell}\\:\n \\widehat Q_{n+\\ell}^{(\\beta)}\\widehat Q_{m-\\ell}^{(-1)}|0\\rangle.\n\\end{equation}\nThese explicit formulas in terms of (\\ref{macQ})\nstrongly suggest that the screening currents\nfor \\v should be\n\\begin{eqnarray}\n S_+(z)\n &\\!\\!=\\!\\!&\n \\exp\\left\\{\\ps{n}\\frac{1-t^n}{1-q^n}\\frac{a_{-n}}{n}z^{n}\\right\\}\n \\exp\\left\\{{\\rm annihilation\\; part\\; for }S_+\n \\right\\} e^{\\beta Q}z^{2\\beta a_0},\\\\\n S_-(z)\n &\\!\\!=\\!\\!&\n \\exp\\left\\{-\\ps{n}\\frac{a_{-n}}{n}z^{n}\\right\\}\n \\exp\\left\\{{\\rm annihilation \\;part \\;for } S_- \\right\\}\n e^{- Q}z^{-2 a_0}.\n\\end{eqnarray}\nNote that this can be regarded as a good deformation of the\nscreening currents (\\ref{Vscr1}) and (\\ref{Vscr2}). \nIt is assumed that \nthe zero-mode parts are not deformed.\n\\\\\n\nIn \\cite{rSKAO}, it is shown that the problem of finding \nall these unknown parts for $\\widehat{D}_{q,t}$, $\\psi_n$ and \n$S^\\pm(z)$ is ``uniquely solved'' if we start from a nice \nansatz for the operators. The reason or mechanism\nbehind the process of solving the problem have not been \nwell investigated yet and seem somehow mysterious. \nWhat is clear so far is that the ``locality'' for \nthe operators seems quite important, {i.e.}, \nthe behaviors of the $\\delta$-functions\nin the OPE factors must be well controlled in some sence.\nThe mathematical structure of this ``locality'' should be \nunderstood in the future.\nTherefore,\nwe will not show the processes of solving this \nproblem here.\nLet us just summarize the results.\n\\proclaim Theorem 1. \\hspace{-2mm}\\cite{rSKAO} \nThe quantum deformed Virasoro current $T(z)$ is realized as\n\\begin{eqnarray}\\label{eq:qVirFFR1}\nT(z)&=& \\Lambda^+(z)+\\Lambda^-(z),\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\Lambda^+(z)\n &\\!\\!\\!\\!\\! = \\!\\!\\!\\!& p^{1\/2}\\exp\\left\\{-\\ps{n}\\frac{1-t^n}{1+p^n}\n\\frac{a_{-n}}{n}z^{n}t^{-n}p^{-n\/2}\\right\\}\n \\exp\\left\\{-\\ps{n}(1-t^n)\\frac{a_{n}}{n}z^{-n}p^{n\/2}\\right\\}\n q^{\\beta a_{0}}, \\nonumber \\\\\n\\Lambda^-(z)\n &=&\\!\\!\\! p^{-1\/2}\\exp\\left\\{\\ps{n}\\frac{1-t^n}{1+p^n}\n\\frac{a_{-n}}{n}z^{n}t^{-n}p^{n\/2}\\right\\}\n \\exp\\left\\{\\ps{n}(1-t^n)\\frac{a_{n}}{n}z^{-n}p^{-n\/2}\\right\\}\n q^{-\\beta a_{0}}.\n\\label{e:b1}\n\\end{eqnarray}\n\n\\noindent By studying the operator product expansions, it is \neasy to see that\nthis $T(z)$ satisfies the relation of \\v (\\ref{e:a1.2}).\nWe can observe that this formula has strong resemblance to the\ndressed vacuum form (DVF) in the algebraic Bethe ansatz.\nThis profound $q$-Virasoro-DVF correspondence \n($T(z)=\\Lambda^+(z)+\\Lambda^-(z)$) was\ndiscovered by Frenkel and Reshetikhin \\cite{rFR}.\n\n\\proclaim Theorem 2. \\hspace{-2mm}\\cite{rSKAO} \nThe Macdonald operator is written as\n\\begin{eqnarray}\n\\widehat{D}_{q,t}&=&\\frac{t^N}{t-1}\\left[\\oint\\frac{dz}{2\\pi\\i}\\frac{1}{z}\n\\psi(z)T(z) -p^{-1}q^{-2\\beta a_0}\\right] -\\frac{1}{t-1}\n\\nonumber \\\\\n&= &\\frac{t^N}{t-1}\\left[\\pzs{n}\\psi_{-n}T_n\n -p^{-1}q^{-2\\beta a_0}\\right]-\\frac{1}{t-1},\n\\label{e:b7.3}\n\\end{eqnarray}\nwhere the field $\\psi(z)$ is given by\n\\begin{equation}\n\\psi(z)=\\pzs{n}\\psi_{-n}z^n=p^{-1\/2}\\exp\\left\\{-\\ps{n}\\frac{1-t^n}{1+p^n}\n\\frac{a_{-n}}{n}z^np^{n\/2}t^{-n}\\right\\}\n q^{-\\beta a_0 }\n\\end{equation}\n\n\\proclaim Theorem 3. \\hspace{-2mm}\\cite{rSKAO} The screening currents for \\v\n\\begin{eqnarray}\n S_+(z)\n &\\!\\!=\\!\\!&\n \\exp\\left\\{\\ps{n}\\frac{1-t^n}{1-q^n}\\frac{a_{-n}}{n}z^{n}\\right\\}\n \\exp\\left\\{-\\ps{n}(1+p^n)\\frac{1-t^n}{1-q^n}\\frac{a_{n}}{n}z^{-n}\n \\right\\} e^{\\beta Q}z^{2\\beta a_0},\n \\label{e:c1.1}\\\\\n S_-(z)\n &\\!\\!=\\!\\!&\n \\exp\\left\\{-\\ps{n}\\frac{a_{-n}}{n}z^{n}\\right\\}\n \\exp\\left\\{\\ps{n}(1+p^n)\\frac{a_{n}}{n}z^{-n}p^{-n}\\right\\}\n e^{- Q}z^{-2 a_0},\n \\label{e:c1.2}\n\\end{eqnarray}\nsatisfies the \ncommutation relation:\n\\begin{eqnarray}\n \\Bigl[T_n,S_+(w)\\Bigr]\n &\\!\\!=\\!\\!&\n -(1-q)(1-t^{-1})\\frac{d_q}{d_q w}\n \\left((p^{-\\frac12}w)^{n+1}p^{\\frac12}:\\Lambda^-(p^{-\\frac12}w)S_+(w):\n \\right), \\label{e:c2.1}\\\\\n \\Bigl[T_n,S_-(w)\\Bigr]\n &\\!\\!=\\!\\!&\n -(1-q^{-1})(1-t)\\frac{d_t}{d_t w}\n \\left((p^{\\frac12}w)^{n+1}p^{-\\frac12}:\\Lambda^+(p^{\\frac12}w)S_-(w):\n \\right),\n \\label{e:c2.2}\n\\end{eqnarray}\nwhere\nthe difference operator with one parameter\nis defined by\n\\begin{equation}\n\\frac{d_\\xi}{d_\\xi z}g(z)=\\frac{g(z)-g(\\xi z)}{(1-\\xi)z}.\n\\label{e:c6}\n\\end{equation}\n\nSince the Kac determinant has the same structure as \nthe ordinary Virasoro algebra has, the structure of the\nsingular vectors are not changed in an essential manner.\nAs for the case of the Jack polynomials, \nwe can write down all the singular vectors at least\nformally in the following way\n\\begin{eqnarray}\\label{eq:qVirSing}\n \\ket{\\chi_{r,s}}\n &\\!\\!=\\!\\!&\n \\oint\\prod_{j=1}^r\\frac{dz_j}{2\\pi i}\\cdot\n \\prod_{i=1}^r S_+(z_i)\n \\ket{\\alpha_{-r,s}} ,\n\\end{eqnarray}\nhowever, we have to be very careful about the \nintegration cycle.\nRecently, very nice construction of \nthe integration cycles together with\nsome modification of the integration kernel\nis achieved in \\cite{rLP2,rJLMP}.\nA $q$-deformation of the Felder complex is \nconstructed in these works, and \nmathematical treatment becomes much ``easier'' than\nthe original case.\n\nFinally, we have as the Jack case,\n\n\\proclaim Theorem 4.\nThere exists a one to one correspondence between \nthe singular vectors $|\\chi_{r,s}\\rangle $ of the $q$-Virasoro algebra \\v and\nthe Macdonald functions $P_{\\{ s^r\\}}(x;q,t)$ \nwith the rectangular Young diagram ${\\{ s^r\\}}$ up to \nnormalization constants. \nIt is simply given by\n\\begin{eqnarray}\nP_{\\{ s^r\\}}(x;q,t) \\propto \n \\langle \\alpha_{r,s}| \n\\exp\\left\\{\\sum_{n=1}^\\infty \\frac{1-t^n}{1-q^n}\\frac{a_n}{n}\np_n\\right\\}|\\chi_{r,s}\\rangle,\n\\end{eqnarray}\nwhere $\\langle \\alpha_{r,s}| \\alpha_{r,s}\\rangle=1$.\n\nFor the general Young diagram, \nthe Macdonald polynomials correspond to\nthe singular vectors of $q$-${\\cal W}$ algebras \\cite{rAKOS}.\n\n\n\n\\subsection{\\v and the Hall-Littlewood polynomial}\n\\label{q0limit}\nIf we take the limit $q \\to 0$, the Macdonald \npolynomial $P_\\lambda(x;q,t)$ reduces to the \nHall-Littlewood polynomial $P_\\lambda(x;t)$ \\cite{rM}.\nLet us study the limit of \\v in $q \\to 0$,\nand the connection \nbetween \\v and the Hall-Littlewood polynomials $P_\\lambda(x;t)$.\nThe commutation relations are already given in (\\ref{r2}). \nThe Kac determinants at lower levels are calculated as\n\\begin{eqnarray}\n \\det{}_1 &\\!\\!=\\!\\!&\n \\langle \\lambda| {\\tilde T}_1 {\\tilde T}_{-1}|\\lambda\\rangle =1-t^{-1}, \\nonumber\\\\ \n \\det{}_2\n &\\!\\!=\\!\\!&\n \\left|\n \\begin{array}{cc}\n \\langle\\lambda| {\\tilde T}_{2}{\\tilde T}_{-2}|\\lambda\\rangle &\n \\langle\\lambda| {\\tilde T}_{2}{\\tilde T}_{-1}{\\tilde\nT}_{-1}|\\lambda\\rangle \\\\\n \\langle\\lambda| {\\tilde T}_{1}{\\tilde T}_{1}{\\tilde T}_{-2}|\\lambda\\rangle &\n \\langle\\lambda| {\\tilde T}_{1}{\\tilde T}_{1}{\\tilde T}_{-1}{\\tilde\nT}_{-1}|\\lambda\\rangle\n \\end{array}\n \\right| \\nonumber\\\\\n &\\!\\!=\\!\\!&\n (1-t^{-1})^2 (1-t^{-2}).\n \\label{r5}\n\\end{eqnarray}\nHere,\nwe observe that the Kac determinants do not depend on $\\lambda$.\nTherefore, if $t$ is generic, we have no singular vectors for any $\\lambda$.\nTo study the degeneration of the boson realization, \nwe have to restrict the zero-mode charge of the vacuum \n$|\\alpha_{r,s}\\rangle$ to $s= 0$, otherwise, we are not able to \nobtain nontrivial algebra.\nIt is easy to see that the operators\n$\\tilde{\\Lambda}^\\pm(z)\\equiv\n\\lim_{q\\rightarrow0}\\Lambda^\\pm(p^{\\pm1\/2}z)$ are well behaving ones.\nIntroducing the renormalized boson\n$b_n=- t^{n} a_n$, $b_{-n}= a_{-n}$ for $n>0$, \nwe have in $q\\rightarrow0$, \n\\begin{equation}\n[b_n,b_m]=n {1 \\over 1-t^{-|n|}}\\delta_{n+m,0},\n\\end{equation}\n\\begin{eqnarray}\n\\tilde{\\Lambda}^+(z)\n &=& t^{r\/2} \\exp\\left\\{\\ps{n} (1-t^{-n})\n\\frac{b_{-n}}{n}z^{n} \\right\\}\n \\exp\\left\\{-\\ps{n}(1-t^{-n})\\frac{b_{n}}{n}z^{-n}\\right\\}, \n\\label{eq:BosonHall} \\\\\n\\tilde{\\Lambda}^-(z)\n &=& t^{-r\/2} \\exp\\left\\{-\\ps{n}(1-t^{-n})\n\\frac{b_{-n}}{n}z^{n}\\right\\}\n \\exp\\left\\{\\ps{n}(1-t^{-n})\\frac{b_{n}}{n}z^{-n}\\right\\},\\nonumber \n\\end{eqnarray}\non the Fock space spanned over $|\\alpha_{r,0}\\rangle$.\nNote that these are essentially the same as \nJing's operators $H(z)$ and $H^*(z)$ for the \nHall-Littlewood polynomial $P_\\lambda(x;t^{-1})$ \\cite{rJ}.\nUsing this notation, the rescaled \\v generator\n$ {\\tilde T }_n$ is expressed as\n\\begin{equation}\n{\\tilde T }_n =\\oint\\frac{dz}{2 \\pi i z}\n\\left(\\,\\, \\theta[\\,n\\leq0\\,] \\tilde{\\Lambda}^+(z) + \n \\theta[\\,n\\geq0\\,] \\tilde{\\Lambda}^-(z)\n\\,\\,\\right) z^n,\n\\end{equation}\nwhere $\\theta[P]=1$ or $0$ \nif the proposition $P$ is true or false, respectively. \nThis formula and the coincidence of our\n$\\tilde{\\Lambda}^+(z)$ with Jing's $H(z)$ means that \nin $q\\rightarrow0$ limit,\n``all the vectors in the Fock module'' are\nwritten in terms of the Hall-Littlewood polynomials.\n\nThe screening currents will disappear in this limit\nin the sense that $S_-(z)$ and\n$\\left[\\tilde{T}_n,S_+(w)\\right]$ become singular.\nThe disappearance of the singular vectors in the Fock module \nwhich is derived from the study of the Kac determinants\nis explained by this singular behavior of the \nscreening currents.\n\nIt seems interesting to study the relation between \\v and the\nHall algebra \\cite{rM} which is related with the analysis over \nthe local fields.\nIs it possible to have a geometric interpretation \nof \\v for $q=0$? \n\n\n\n\\section{Further aspects of \\v and relations with other models}\n\n\nHere we discuss {\\it i)}\nthe representation theories, {\\it ii)} application to the ABF model and \n{\\it iii)} a hidden elliptic algebra generated by screening current.\nWe will also study the limits of $\\beta=1$, $3\/2$ and $2$,\nand find some connections with the Kac-Moody algebras\nwhen $\\beta=1$, for $\\beta=3\/2$, the $q$-Virasoro generator\nis given by a BRST exact form and construct a topological model,\nand when $\\beta=2$, the $q$-Virasoro algebra relates\nwith $c=1$ ${\\cal W}_{1+\\infty}$ algebra. \n\n\n\\subsection{symmetric realization of \\v and vertex operators}\n\\label{secSYMMETRIC}\n\nWhen we introduced the Feigin-Fuchs realization,\nthe creation operators and the annihilation operators\nhad nice symmetry. However we destroyed this symmetry \nto make many formulas for the Jack and Macdonald \nsymmetric polynomials become simple.\nWe will study some applications of \\v to other \nsolvable systems. So, it helps us very much\nto have a ``symmetric expressions'' of \n$T(z)$ and $S_\\pm(z)$ \\cite{rFFr,rAKOS,rLP2,rKa,rAKMOS}.\n\nLet us introduce the fundamental Heisenberg algebra\n$h_n$ ($n\\in{\\bf Z}$), $Q_{h}$ having the commutation relations\n\\begin{eqnarray}\n [h_n,h_m]\n &\\!\\!=\\!\\!&\n \\frac{1}{n}\\frac{(q^{\\frac{n}{2}}-q^{-\\frac{n}{2}})\n (t^{\\frac{n}{2}}-t^{-\\frac{n}{2}})}{p^{\\frac{n}{2}}+p^{-\\frac{n}{2}}}\n \\delta_{n+m,0}\n ,\\qquad\n [h_n,Q_h]=\\frac12\\delta_{n,0}. \\label{boscom}\n\\end{eqnarray}\nThe correspondence to the bosonic oscillators \nin the last two subsections is $(n>0)$\n\\begin{equation}\nh_{ n}={t^n -1 \\over n} a_{ n},\\qquad\nh_{-n}= {1\\over n}{1-t^{-n}\\over 1+p^n} a_{-n},\\qquad\nh_0 = \\sqrt\\beta a_0,\\qquad\nQ_h ={\\sqrt\\beta\\over 2} Q.\n\\end{equation}\nBy these, \nthe Virasoro current $T(z)$ and the screening current $S_{\\pm}(z)$\n(with some modification) are written as\n\\begin{eqnarray}\n T(z) &\\!\\!\\!=\\!\\!\\!& \\Lambda^+(z)+\\Lambda^-(z),\n\\label{eq:qVirFFR2}\\\\\n \\Lambda^\\pm(z) &\\!\\!\\!=\\!\\!\\!&\n :\\exp\\left\\{\\pm\\sum_{n\\neq 0}h_np^{\\pm\\frac{n}{2}}z^{-n}\\right\\}:\n q^{\\pm\\sqrt{\\beta}h_0}p^{\\pm\\frac12}, \\\\\n S_{+}(z) &\\!\\!\\!=\\!\\!\\!& \n :\\exp\\left\\{- \\sum_{n\\neq 0}\n \\frac{p^{\\frac{n}{2}}+p^{-\\frac{n}{2}}}\n {q^{\\frac{n}{2}}-q^{-\\frac{n}{2}}}\n h_nz^{-n}\\right\\}:\n e^{2\\sqrt{\\beta}Q_h}z^{2\\sqrt{\\beta}h_0},\n\\label{eq:SC21}\\\\\n S_{-}(z) &\\!\\!\\!=\\!\\!\\!& \n :\\exp\\left\\{ \\sum_{n\\neq 0}\n \\frac{p^{\\frac{n}{2}}+p^{-\\frac{n}{2}}}\n {t^{\\frac{n}{2}}-t^{-\\frac{n}{2}}}\n h_nz^{-n}\\right\\}:\n e^{-{2\\over\\sqrt{\\beta}}Q_h}z^{-{2\\over\\sqrt{\\beta}}h_0}.\n\\label{eq:SC22}\n\\end{eqnarray}\nIf we introduce the isomorphisms $\\theta$ and $\\omega$ \nof the Heisenberg algebra\nrelated with (\\ref{e:aa1}), (\\ref{e:aa2}):\n\\begin{eqnarray}\n &&{\\rm (II')}\\;\\;\\;\\; \\theta:\n (q,t) \\mapsto (q^{-1},t^{-1}),\\quad \n h_n \\mapsto -h_n \\:(n\\neq 0),\n\\quad\n h_0 \\mapsto h_0, \\quad Q_h \\mapsto Q_h, \\cr\n && {\\rm (III')} \\;\\;\\;\\;\\omega: \\quad \n q \\leftrightarrow t \\quad, \\quad \n h_n \\mapsto -h_n,\n \\quad Q_h \\mapsto - Q_h , \\label{symm}\n\\end{eqnarray}\nthen\n$\\Lambda^-(z) = \\theta\\cdot \\Lambda^+(z) = \\omega\\cdot \\Lambda^+(z)$,\n$S_{-}(z) = \\omega\\cdot S_{+}(z)$ and\n$ \\theta\\cdot S_{\\pm}(z)= S_{\\pm}(z)$.\nHere $\\omega\\cdot \\beta$ should be understood as $1\/\\beta$.\nUnder the isomorphism $\\sigma$ such that:\n\\begin{equation}\n {\\rm (IV)}\\;\\;\\;\\; \\sigma:\n q \\leftrightarrow 1\/t, \\qquad \n \\sqrt\\beta \\leftrightarrow -\\sqrt{1\/\\beta},\n\\end{equation}\n$\\sigma\\cdot\\Lambda^\\pm(z) = \\Lambda^\\pm(z)$ and\n$\\sigma\\cdot S_\\pm(z) = S_\\mp(z)$.\n\n\nThe free boson realization for $T(z)$ is expressed as \nthe following deformed Miura transformation \\cite{rFR}\n\\begin{equation}\n:\\!\\left( p^D - \\Lambda^+(z) \\right) \\left( p^D - \\Lambda^-(z) \\right)\\!:\\,\n= p^{2D} - T(z) p^D + 1,\n\\label{eq:qMiura}\n\\end{equation}\nwhich has been generalized to define \nthe $q$-deformed $\\cal W$ algebra \\cite{rFFr,rAKOS}.\nBy using this transformation,\nFrenkel-Reshetikhin \\cite{rFR} proposed a generalization of\ntheir quasi-classical $q$-Virasoro algebra to $ABCD$-type cases.\nAn analogy to the Baxter's dressed vacuum form $Q$ defined by\n$:\\!\\left( p^D - \\Lambda^-(z) \\right) Q(z)\\!:\\, = 0$,\nso $\\Lambda^\\pm(z) = \\,:\\! Q(zp^{\\mp1})\\, Q(z)^{-1}\\!:$,\nseems to be of some interest.\n\n\nThe vertex operator defined by\n\\begin{equation}\nV_{2,1}(z) \\equiv \\,:\\! \\exp\\left\\{\n\\sum_{n\\neq0}{h_n\\over q^{n\\over2} - q^{-{n\\over2}} } z^{-n} \\right\\}\\!:\ne^{-\\sqrt\\beta Q_h} z^{-\\sqrt\\beta h_0},\n\\label{eq:Vertex21}\n\\end{equation}\nsatisfies\n$\\theta\\cdot V_{2,1}(z) = V_{2,1}(z)$ and\n\\begin{eqnarray}\ng\\left({w\\over z}p^{\\pm{1\\over2}}\\right) T(z) V_{2,1}(w) \n&\\!\\!\\!-\\!\\!\\!&\nV_{2,1}(w) T(z) g^{-1}\\left({z\\over w}p^{\\mp{1\\over2}}\\right)\\\\\n&\\!\\!\\!=\\!\\!\\!&\nt^{\\pm{1\\over4}}(t^{1\\over2} - t^{-{1\\over2}})\\,\n\\delta\\left(t^{\\pm{1\\over2}}{w\\over z}\\right)\n V_{2,1}(q^{\\pm{1\\over2}} w) p^{\\mp{1\\over2}},\n\\label{eq:T&V21}\n\\end{eqnarray}\nwith\n\\begin{equation}\ng(x) = \nt^{\\pm{1\\over4}}\n\\exp\\left\\{\\pm\\sum_{n>0}{1\\over n}\n{ t^{n\\over2} - t^{-{n\\over2}} \\over p^{n\\over2} - p^{-{n\\over2}} }x^n\n\\right\\}.\n\\end{equation}\nNote that\n$ V_{2,1}(q^{\\pm{1\\over2}} w) p^{\\mp{1\\over2}} =\n\\,:\\! \\Lambda^\\mp(t^{\\mp{1\\over2}}w) V_{2,1}(w)\\!:\\,$. \nIf we let\n$V_{1,2}(z) \\equiv \\sigma\\cdot V_{2,1}(z)$ and their fusion as\n\\begin{equation}\nV_{\\ell+1,k+1}(z) \\equiv \\,:\\!\n\\prod_{i=1}^\\ell V_{2,1}(q^{\\ell+1-2i\\over2\\ell}z)\n\\prod_{j=1}^k V_{1,2}(t^{k +1-2j\\over2k }z)\n\\!:,\n\\label{eq:VertexLK}\n\\end{equation}\nthen they also obey a similar commutation relation as \\eq{eq:T&V21}, \nwhich reduces to the usual defining relation for the Virasoro primary field\nof the conformal weight $h_{\\ell+1, k+1}$,\nin the limit $q\\rightarrow1$ \\cite{rAKMOS}.\nThe adjont action of the \\v generator $T(z)$ \non this fused vertex operator $V_{\\ell+1,k+1}$\nmay be closely connected with \na coproduct of the algebra \\v.\nSimilar but slightly different definition for fused vertex operators\n\\begin{equation}\n :\\!\n \\prod_{i=1}^{\\ell}V_{2,1}(q^{-\\frac{k}{2}}t^{\\frac{\\ell+1}{2}-i}z)\n \\prod_{j=1}^k V_{1,2}(t^{-\\frac{\\ell}{2}}q^{\\frac{k+1}{2}-j}z)\n \\!:, \\label{kade}\n\\end{equation}\nwas proposed in \\cite{rKa}. \nThe meaning of fused operators given by (\\ref{eq:VertexLK}) or\n(\\ref{kade}) has not been made clear yet. \n\n\n\nThe fundamental vertex operators $V_{2,1}(z)$ and $V_{1,2}(z)$,\nthat satisfy fermion like anti-commutation relation,\nare especially important.\nBecause\nthe $q$-Virasoro generator and screening currents are expressed by them\nas follows;\n\\begin{equation}\n \\Lambda^+(zp^{1\\over 2}) \n=\\,\\, : V_{2,1}^+(zq^{-{1\\over 2}}) V_{2,1}^{-}(zq^{1\\over 2}) : \np^{1\\over 2},\\qquad\n S^+(z) =\\,\\, : V_{2,1}^{-}(zp^{1\\over 2}) V_{2,1}^{-}(zp^{-{1\\over 2}}) :,\n\\label{eq:TSbyV21}\n\\end{equation}\nand the relations obtained by $\\omega$.\nHere\n$V^\\pm_{\\ell+1,k+1}(z)\\equiv V^{\\pm1}_{\\ell+1,k+1}(z)$.\nMoreover, \nthe boson and power-sum correspondence operator in eq.\\ \\eq{eq:BosonMacOp}\nis also realized as $:\\prod_{i=1}^N V_{2,1}(q^{1\\over2} x_i^{-1}):$.\nHence, they must play more important role in the $q$-Virasoro algebra.\n\n\n\n\\subsection{ABF model and \\v}\n\n\nIn the papers \\cite{rLP2}, the \nexplicit formula for the multipoint \ncorrelation functions is successfully obtained. \nWe review their method and the relation to \\v.\n\nThe $q$-Virasoro algebra can be applied to the off-critical phenomena,\nespecially to the ABF model in the regime \n\\uppercase\\expandafter{\\romannumeral3} \\cite{rABF}.\nLet $z\\equiv p^v$ and the vertex operators $\\Phi_\\pm(z)$ be\n\\begin{equation}\n\\Phi_+(z)\\equiv V_{2,1}(z),\\qquad\n\\Phi_-(z') \\equiv \n \\oint{dz\\over 2\\pi iz} V_{2,1}(z') S_+(z) z^\\beta f(v-v',\\pi),\n\\label{e:ABFVertex}\n\\end{equation}\nwhere\n$[v] \\equiv p^{{1\\over2} ((1-\\beta)v^2-v) }\n(z;q)_\\infty (qz^{-1};q)_\\infty (q;q)_\\infty $\nand \n\\begin{equation}\nf(v,w)\\equiv\n{[v+{1\\over2}-w]\\over[v-{1\\over2}]},\\qquad\n\\pi\\equiv\n-{2h_0 \\over \\sqrt\\beta-\\sqrt{1\/\\beta}}.\n\\end{equation}\nThe integration contour is a closed curve around the origin satisfying \n$p|z'|<|z|0}|\\RR,\\SSS\\rangle = 0,\\qquad\nh_0 |\\RR,\\SSS\\rangle = \n-{1\\over2}\\left( \\RR\\sqrt\\beta - \\SSS\\sqrt{1\/\\beta} \\right)|\\RR,\\SSS\\rangle.\n\\end{equation}\nSuppose \n$\\beta = \\QQ\/\\PP$ with coprime integers $\\PP>\\QQ\\in{\\bf N}$ and\nlet the screening charge \n$\\SC_+\\,:\\,\\cF{\\SSS}{\\RR}\\rightarrow\\cF{\\SSS}{\\RR-2}$ be\n\\begin{equation}\n\\SC_+= \\oint{dz\\over 2\\pi iz} S_+(z) z^\\beta f(v,\\pi),\n\\end{equation}\nand define the BRST charges $Q^+_j$ ($j\\in{\\bf Z}$) as\n\\begin{eqnarray}\nQ^+_{2j } =\n\\SC_+^\\RR \\,\\,\\,\\,&:&\n\\cF{\\SSS}{ \\RR-2j\\PP}\\,\\,\\,\\rightarrow\\,\\,\\cF{\\SSS}{-\\RR-2 j \\PP},\\cr\nQ^+_{2j+1} =\n\\SC_+^{\\PP-\\RR}\\!\\!\\!\\!&:&\n\\cF{\\SSS}{-\\RR-2j\\PP} \\rightarrow\\,\\,\\cF{\\SSS}{ \\RR-2(j+1)\\PP}.\n\\label{eq:BRST}\n\\end{eqnarray}\nWe also define the dual screening charge\n$\\SC_-\\,:\\,\\cF{\\SSS}{\\RR}\\rightarrow\\cF{\\SSS-2}{\\RR}$ and\nthe dual BRST charges $Q^-_j$\nby the replacement \n$\\sqrt\\beta\\leftrightarrow -\\sqrt{1\/\\beta}$, \n$q\\leftrightarrow 1\/t$ and\n$\\RR\\leftrightarrow \\SSS$.\n\n\n\\proclaim Proposition 4. \\hspace{-2mm}\\cite{rLP2,rJLMP}\nThe screening charges $\\SC_\\pm$ commute with each other and \nwith $q$-Virasoro generators\n\\begin{eqnarray}\n[\\SC_+,\\SC_-] \n&\\!\\!\\!=\\!\\!\\!&\n0,\\cr\n[T(z), \\SC_\\pm^{n_\\pm}]\n&\\!\\!\\!=\\!\\!\\!&\n0, \\quad \n{\\rm on}\\,\\,\\,\\, \\cF{\\RR_-}{\\RR_+},\\quad\n{\\rm with}\\,\\,\\,\\, n_\\pm\\equiv \\RR_\\pm\\,\\, {\\rm mod} \\,\\,P_\\pm,\n\\end{eqnarray}\nand are also nilpotent\n\\begin{equation}\nQ^\\pm_j Q^\\pm_{j-1} \n=\n\\SC_\\pm^{P_\\pm} = 0, \\quad P_\\pm>1.\n\\label{eq:nilpotent}\n\\end{equation}\n\nHence we can construct Felder type BRST complexes, \nfor example, by $\\SC_+$\n\\begin{equation}\n\\cdots\n\\BRS{\\SC_+^{ \\RR}}\\cF{\\SSS}{-\\RR+2\\PP}\n\\BRS{\\SC_+^{\\PP-\\RR}}\\cF{\\SSS}{ \\RR}\n\\BRS{\\SC_+^{ \\RR}}\\cF{\\SSS}{-\\RR}\n\\BRS{\\SC_+^{\\PP-\\RR}}\\cF{\\SSS}{ \\RR-2\\PP}\n\\BRS{\\SC_+^{ \\RR}}\n\\cdots.\n\\label{eq:Felder}\n\\end{equation}\n{}From the Kac determinant in (\\ref{Kacconj}),\nthe Fock module $\\cF{\\SSS}{\\RR}$ with $\\RR$, $\\SSS\\in{\\bf N}$ is reducible. \nTo obtain an irreducible one $\\cL{\\SSS}{\\RR}$, \nwe have to factor out the submodules by the Felder resolution.\nIn a special case, this irreducible module coincides with \nthe space of the ABF model.\n\n\nTo see this, we have to introduce a grading operator,\nwhich plays the role of the corner Hamiltonian in the ABF model,\n\\begin{equation}\nH_c = \\sum_{n>0} n^2 \n {p^{n\\over2}+p^{-{n\\over2}}\\over\n\\left( q^{n\\over2}-q^{-{n\\over2}}\\right) \n\\left( t^{n\\over2}-t^{-{n\\over2}}\\right) }\nh_{-n} h_n + h_0^2 -{1\\over 24}.\n\\label{eq:cornerHamil}\n\\end{equation}\nThis commutes with screening currents up to a total divergence\n\\begin{equation}\n[H_c,S_\\pm(z)] z^{\\beta^{\\pm1}} \n= {\\partial\\over\\partial z} \\left( S_\\pm(z) z^{\\beta^{\\pm1}}\\right),\n\\end{equation}\nand its eigenvalues $\\EPS{\\SSS}{\\RR}$ \non the Fock module $\\cF{\\SSS}{\\RR}$ are\n\\begin{equation}\n\\EPS{\\SSS}{\\RR}=\\DEL{\\SSS}{\\RR} - {c\\over 24} + n,\\qquad \nn\\in{\\bf Z}_{n\\geq0}\n\\end{equation}\nwith $\\DEL{\\SSS}{\\RR}$ in eq.\\ \\eq{eq:KacFormula}.\nWhen $\\QQ=\\PP-1$,\nthese values coincide with the eigenvalues of \nthe corner Hamiltonian of the ABF model\ncorresponding to the $1$-$d$ configurations given by the rule:\n{\\it i}) each height takes an integer value between \n$1$ and $\\PP-1$, {\\it ii}) the allowed values of difference\nin any neighboring heights are $\\pm1$, {\\it iii})\nthe height at the origin is $\\RR$,\n {\\it iv}) the asymptotic configuration is \n$\\cdots,\\SSS,\\SSS+1,\\SSS,\\SSS+1,\\cdots$.\nHowever, we should note that the \nmultiplicities of the bosonic Fock space and \nthat of ABF model are different.\n\nLukyanov and Pugai \\cite{rLP1,rLP2} showed that,\nafter the Felder-type BRST resolution by the dual screening current $S_-(z)$,\nthe multiplicities in the irreducible Fock module $\\cL{\\SSS}{\\RR}$ \ncoincide those of the ABF model.\nTherefore, ABF model is completely described by \nthe representation of the $q$-Virasoro algebra\nand the multi-point local height probabilities of ABF model \\cite{rFJMMN} are\nrealized as correlation functions of the vertex operators.\nFor example, the probability that \nthe heights at the same vertical column sites have the values \n$1\\leq\\RR_1,\\RR_2, \\cdots,\\RR_n\\leq\\PP-1$\nis proportional to\n\\begin{equation}\nTr_{\\cL{\\SSS}{\\RR_1}}\n\\left[ p^{2H_c}\n\\Phi_{-\\sigma_1 }(z_1 \/p)\\cdots\n\\Phi_{-\\sigma_{n-1}}(z_{n-1}\/p)\n\\Phi_{ \\sigma_{n-1}}(z_{n-1} )\\cdots\n\\Phi_{ \\sigma_1 }(z_1 )\n\\right],\n\\end{equation}\nwhere \n$\\sigma_s = \\RR_{s+1} - \\RR_s$.\n\n\n\n\n\\subsection{elliptic algebra generated by the screening currents and \n$k=1$ affine Lie algebra}\n\n\nThe properties of screening currents are quite important \nin the representation theory of the infinite-dimensional algebra;\nthey govern the irreducibility and the physical states\nas mentioned above subsections.\nMoreover they relate with hidden quantum symmetries.\n\n\n\\subsubsection{an elliptic algebra generated by $S^\\pm(z)$}\n\n\nHere, we show that the\nscreening currents generate an elliptic hidden symmetry,\nwhich reduces to the (quantum) affine Lie algebra with a special center\nwhen $c$ tends to $1$.\nLet us introduce a new current \n$\\Psi(z) \\equiv \\,:\\!S_+(q^{\\pm{1\\over2}}z)\\, S_-(t^{\\pm{1\\over2}}z)\\!:$, \n{\\it i.e.},\n\\begin{equation}\n\\Psi(z)\n= \\exp\\left\\{ \\sum_{n\\neq0} \n{ p^n-p^{-n} \\over (\\QINT qn)(\\QINT tn) } h_n z^{-n} \\right\\}\ne^{2\\alpha Q} z^{2\\alpha h_0},\n\\end{equation}\nwith $\\alpha = \\sqrt\\beta - 1\/\\sqrt\\beta$, \nthen we have\n\n\\proclaim Proposition 5.\nScreening Currents $S_\\pm(z)$ and $\\Psi(z)$ generate \nthe following elliptic two-parameter algebra;\n\\begin{eqnarray}\nf_{00}\\left({w\\over z}\\right) \\Psi(z) \\Psi(w) \n&\\!\\!\\!=\\!\\!\\!&\n\\Psi(w) \\Psi(z) f_{00}\\left({z\\over w}\\right),\n\\label{eq:ElliPP}\\\\\nf_{0\\pm}\\left({w\\over z}\\right) \\Psi(z) S_\\pm(w) \n&\\!\\!\\!=\\!\\!\\!&\nS_\\pm(w) \\Psi(z) f_{\\pm0}\\left({z\\over w}\\right),\n\\label{eq:ElliPS}\\\\\nf_{\\pm\\pm}\\left({w\\over z}\\right) S_\\pm(z) S_\\pm(w) \n&\\!\\!\\!=\\!\\!\\!&\nS_\\pm(w) S_\\pm(z) f_{\\pm\\pm}\\left({z\\over w}\\right),\n\\label{eq:ElliSS1}\n\\end{eqnarray}\n\\begin{equation}\n\\left[\\,S_+(z), S_-(w)\\,\\right]\n=\n{1\\over (p-1)w}\n\\left[\\, \n\\delta\\left(p^{ 1\\over2 }{w\\over z}\\right) \\Psi(t^{-{1\\over2}}w) -\n\\delta\\left(p^{-{1\\over2}}{w\\over z}\\right) \\Psi(q^{-{1\\over2}}w)\n\\,\\right],\n\\label{eq:ElliSS2}\n\\end{equation}\nwhere\n$f_{00}(x) =\nf_{++}(x) f_{+-}^2(xp^{1\\over2}) f_{--}(x)$ and \n$f_{0\\pm}(x) =\nf_{\\pm0}(x) =\nf_{+\\pm}(xq^{1\\over2}) f_{-\\pm}(xt^{1\\over2})$ \nwith\n\\begin{eqnarray}\nf_{+-}(x) \n=\nf_{-+}(x) \n&\\!\\!\\!=\\!\\!\\!&\n\\exp\\left\\{ -\\sum_{n>0}{1\\over n}(\\PINT pn)x^n \\right\\} x^{-1},\\\\\nf_{++}(x) \n&\\!\\!\\!=\\!\\!\\!&\n\\exp\n\\left\\{ -\\sum_{n>0}{1\\over n}{\\QINT tn \\over \\QINT qn} (\\PINT pn)x^n \\right\\} \nx^\\beta,\n\\end{eqnarray}\nand $f_{--}(x) = \\omega\\cdot f_{++}(x)$.\\\\\n\\indent\nThe relation between $\\Psi(z)$ and \nthe $q$-Virasoro generators $\\Lambda^\\pm(z)$ \nis simply written as \n\\begin{equation}\n[\\,\\Lambda^\\pm(z),\\Psi(w)\\,] =\n\\mp p^{\\mp{1\\over 2}} (\\QINT p1)\n\\delta\\left({w\\over z}\\right)\\,:\\!\\Lambda^\\pm(w)\\Psi(w)\\!:.\n\\end{equation}\n\n\n\nAs we shall see explicitly in the next subsection,\nin the limit of\n$q$ and $t$ tend to $0$ with $p$ and $t^{-{|n|\\over2}} h_n$ fixed,\nthe relations \\eq{eq:ElliPP}--\\eq{eq:ElliSS2}\nreduce to those of $k=1$ $U_q(\\widehat{sl}_2)$.\nTherefore, the algebra generated by \n$S^\\pm(z)$ and $\\Psi(z)$ can be regarded \nas an elliptic generalization of $U_q(\\widehat{sl}_2)$ with level-one.\nWe can regard $p$, $q$ and $t$ \nas three independent parameters. \nEven in this case, \nscreening currents \\eq{eq:SC21} and \\eq{eq:SC22} and new currents \n$\\Psi_\\pm(z) \\equiv \n\\,:\\!S_+(p^{\\pm{1\\over4}}z)\\, S_-(p^{\\mp{1\\over4}}z)\\!:$ \ngenerate an elliptic algebra. \nThese extended algebras may help us to investigate \nelliptic-type integrable models.\n\n\nIn the sense of analytic continuation,\nthese relations are also rewritten by \nusing elliptic theta functions\\cite{rFFr},\n\\begin{equation}\nS_\\pm(z) S_\\pm(w) \n=\nU_\\pm\\left({w\\over z}\\right) \nS_\\pm(w) S_\\pm(z)\n\\label{eq:ElliSS12}\n\\end{equation}\nwith\n\\begin{equation}%\nU_\\pm(x) =\n-x^{1-2\\beta}\n\\exp\\left\\{ \\sum_{n\\neq0}{1\\over n} \n{ q^{n\\over2} t^{-n} -q^{-{n\\over2}} t^n \\over q^{n\\over2}-q^{-{n\\over2}} }\nx^n \\right\\}\n=\n-x^{2(1-\\beta)} {\\vartheta_1(px;q)\\over\\vartheta_1(px^{-1};q)}, \n\\end{equation}\nand $U_-(x) = \\omega\\cdot U_+(x)$.\nNote that $U_\\pm(x)$ are quasi-periodic functions, namely for\n$U_+(x)$, we have\n\\begin{equation}\nU_+(qx) = U_+(x), \\qquad\nU_+(e^{2\\pi i}x) = e^{-4\\pi i\\beta} U_+(x).\n\\end{equation} \n\nIt should be noted that \nthe screening currents of $q$-${\\cal W}$ \nalgebra \\cite{rFFr,rAKOS} and\n$U_q(\\widehat{sl}_N)$ \\cite{rAOS} also obey similar elliptic relations.\n\n\n\n\\subsubsection{two $c=1$ limits}\n\n\nIt is possible to consider two different $c=1$ limits of \\v.\nThey are related with the (quantum) affine algebra of\n$A_1^{(1)}$-type.\n\\\\\n\n\\noindent(A) Let us consider the limit,\n$q \\to 0 $, $t \\to 0$, $p$ is fixed. \nIn this limit we must have $\\beta \\to 1$.\nthus, this is a ``$c=1$'' limit (see (\\ref{e:a9})).\nWe will see that the screening currents $S_\\pm(z)$ \nreduces to \nthe Frenkel-Jing realization of $U_{q}(\\widehat{ sl}_2)$ at \nlevel-one \\cite{rFJ}.\n\nIntroducing rescaled bosons as \n\\begin{eqnarray}\na_n =-h_n q^{|n|\/2}p^{-|n|\/4} [2n]\\qquad (n\\neq 0),\\qquad \na_0=-2 h_0,\\qquad\nQ = -2 Q_h,\n\\end{eqnarray}\nwe obtain\n\\begin{eqnarray}\n&&[a_n , a_m]= \\frac{[2n][n]}{n} \\delta_{n+m,0} , \\quad \n[a_n , Q]= 2\\delta_{n,0}, \\\\\n&&{S}_{-}(z) \\rightarrow\\exp\\left\\{\\sum_{n=1}^{\\infty} \n\\frac{1}{[n]} a_{-n} z^n p^{-\\frac{n}{4}}\\right\\}\n\\exp\\left\\{-\\sum_{n=1}^{\\infty} \n\\frac{1}{[n]} a_n z^{-n} p^{-\\frac{n}{4}}\\right\\}\ne^{Q}z^{a_0}, \n\\cr\n&&{S}_{+}(z)\\rightarrow \\exp\\left\\{-\\sum_{n=1}^{\\infty} \n\\frac{1}{[n]} a_{-n} z^n p^{\\frac{n}{4}}\\right\\}\n\\exp\\left\\{\\sum_{n=1}^{\\infty} \n\\frac{1}{[n]} a_n z^{-n} p^{\\frac{n}{4}}\\right\\}\ne^{-Q}z^{-a_0},\n\\label{eq:FJ}\n\\end{eqnarray}\nwhere $[n]=({p^{\\frac{n}{2}}-p^{-\\frac{n}{2}}})\/\n({p^{\\frac{1}{2}}-p^{-\\frac{1}{2}}})$. After \nreplacing $p^{\\frac{1}{2}}\\to q$,\nwe can identify screening currents ${S}_{\\pm}$ \nwith the Frenkel-Jing realization of the \nDrinfeld currents of $U_{q}(\\widehat{ sl}_2)$.\n\nIt is quite unfortunate that at this limit,\nthe \\v current becomes singular. Namely,\nit seems difficult to extract nontrivial\nobject from $T(z)$ at this limit. Thus it is still a challenging\nproblem to find a Sugawara construction for $U_{q}(\\widehat{ sl}_2)$.\nThe limit discussed in the next paragraph (B)\nmay be relevant to this problem.\n\\\\\n\n\n\\noindent(B)\nNext, we consider the limit of $\\beta \\to 1$ with $q$ fixed.\nThis is another ``$c=1$'' limit.\nIn this limit, the screening currents degenerate to \nthe Frenkel-Kac realization of the level-one $\\widehat{sl}_2$ \\cite{rFK}.\n\nIntroducing rescaled bosons as \n\\begin{eqnarray}\n a_n = \\frac{ 2 n}{q^{\\frac{n}{2}}-q^{-\\frac{n}{2}}} h_n \\quad\n (n \\neq 0), \\quad \n a_0 = 2h_0, \\quad\n Q = 2 Q_h,\n\\end{eqnarray}\nwe obtain\n\\begin{eqnarray}\n&&[a_n , a_m]= 2 n \\delta_{n+m,0} , \\quad \n[a_n , Q]= 2\\delta_{n,0}, \\\\\n&&{S}_{\\pm}(z) \\rightarrow\\exp\\left\\{ \\pm \\sum_{n\\neq 0}\n\\frac{a_{n}}{n} z^{-n} \\right\\}\ne^{\\pm Q}z^{\\pm a_0}.\n\\end{eqnarray}\n{}From these, it can be seen that the screening currents have\nreduced to the Frenkel-Kac realization of $\\widehat{sl}_2$.\n\nIn this limit, \\v survives and satisfies the relation\n\\begin{eqnarray}\n&&\\frac{\\sqrt{(1-q{w \\over z})(1-q^{-1}{w \\over z})}}{1-{w \\over z}}T(z)T(w)-\nT(w)T(z)\\frac{\\sqrt{(1-q{z \\over w})(1-q^{-1}{z \\over w})}}\n{1-{z \\over w}}\\\\\n&=&2(1-q)(1-q^{-1})\n{w \\over z} \\delta'\\left({w \\over z} \\right).\\nonumber\n\\end{eqnarray}\n\nIt is shown that all the vectors in the Fock space\nspanned over the vacuum $|0\\rangle$ are singular vectors of \nthis algebra\nby studying the Kac determinant at level-one.\nThis fact and the normalization of the \nbosonic oscillators in this limit mean that the \nthe Fock space is \nspanned by the Schur symmetric polynomials.\nThis degeneration is not accidental because the Macdonald \npolynomial $P_\\lambda(x;q,t)$ reduces to the \nSchur polynomial in the limit $t\\rightarrow q$.\n\nIn this ``$c=1$'' limit, \\v survives and acts on the Fock space\non which the bosonized currents of $U_q(\\widehat{sl}_2)$ can\nalso act. The relationship between these algebras\nhas not been made clear yet.\n\\\\\n\n\n\n\n\\subsection{$c=0$ topological $q$-superconformal model}\n\n\nThe $c=0$ Virasoro algebra\ndescribes a topological conformal model. \nWe shall call one of the screening currents of this model $G^+(z)$,\nand set $G^-(z)=:(G^+(z))^{-1}:$.\nThe screening charge $\\oint dz G^+(z)$ plays the role of the BRST\noperator.\nSince the model is topological, \nthe energy-momentum tensor $L_{c=0}(z)$ should be BRST \nexact. Actually, we have \n$\\left\\{ \\oint dz' G^+(z') , G^-(z) \\right\\} = 2 L_{c=0}(z)$.\n\nThere exists a similar structure in the $q$-Virasoro case\\footnote{\nThis was inspired by the discussion with T.~Kawai}.\nLet us consider the case when $c=0$, {\\it i.e.}, $\\beta=3\/2$ ($q=p^{-2}$).\nDenote one of the screening current $S_+(z)$ and \nthe normal ordering of its inverse\nas $G^+(z)$ and $G^-(z)$, respectively\n\\begin{equation}\nG^\\pm(z) =\n\\,:\\exp\\left\\{\n\\pm\\sum_{n\\neq0}{h_n \\over p^{n\\over 2}-p^{-{n\\over 2}}} z^{-n} \n \\right\\}:\\, \ne^{\\pm2\\sqrt\\beta Q}\nz^{\\pm2\\sqrt\\beta h_0}.\n\\end{equation}\n\\proclaim Proposition 6. The fields $T(z)$ and $G^\\pm(z)$ satisfy \nthe relations\n\\begin{eqnarray}\nf\\left( {w\\over z}\\right) T(z) T(w) - T(w) T(z) f\\left( {z\\over w}\\right) \n&\\!\\!\\! = \\!\\!\\!& \n-(p^{1\\over 2}+p^{-{1\\over 2}})(p^{3\\over 2}-p^{-{3\\over 2}}) \n\\left( \\delta\\left( {wp\\over z}\\right) \n- \\delta\\left( {w\\over zp}\\right) \\right), \\nonumber \\\\\nf\\left( {w\\over z}\\right) T(z) G^-(w) - G^-(w) T(z) f\\left( {z\\over w}\\right) \n&\\!\\!\\! = \\!\\!\\!& G^-(z) (p^{3\\over 2}-p^{-{3\\over 2}}) \n\\left( p^{2}\\delta\\left( {wp\\over z}\\right) \n-p^{-2}\\delta\\left( {w\\over zp}\\right)\\right), \\nonumber \\\\\n\\left\\{\\oint dz G^+(z), G^-(w) \\right\\} &\\!\\!\\! = \\!\\!\\!& \n{p^{-1}\\over w^2 (p-p^{-1})(p^{1\\over 2}-p^{-{1\\over 2}})}\n\\left( T(w) - (p^{1\\over 2}+p^{-{1\\over 2}}) \\right), \\nonumber\\\\\n\\left[ T(z), \\oint dw G^+(w) \\right] \n&\\!\\!\\! = \\!\\!\\!& 0 , \\qquad\\quad\n\\left\\{ G^\\pm(z),G^\\pm(w)\\right\\} = 0.\n\\label{eq:qTopological}\n\\end{eqnarray}\nwhere $f(x)$ is given by (\\ref{structure}) with $q=p^{-2}$.\n\n\\noindent\nNote that $G^-(z)$ is a primary field and\nits commutation relation with $T(z)$ is given by the same function $f(x)$\nin the defining relation of the $q$-Virasoro algebra.\n\nWe can regard these relations as a $c=0$ topological $q$-Virasoro algebra.\nThe screening charge $\\oint dz G_+(z)$ may play the role of BRST operator\nwhich reduces the bosonic Fock space to \nirreducible representation space of the $q$-Virasoro algebra.\nAt the value of the coupling constant $\\beta=3\/2$,\nthe central charge vanishes and \nthe entire Fock space contains only BRST trivial states,\nexcept for the vacuum state. \nThe $q$-Virasoro generator itself (up to a constant)\nis given by a BRST exact form.\nThus the $\\beta=3\/2$ $q$-Virasoro algebra is a topological field theory\nsame as $q=1$ case.\n\n\nThe relations between the currents $G^\\pm(z)$ and $\\Lambda^\\pm(z)$ are\nwritten as\n\\begin{eqnarray*}\nf\\left( {w\\over z}\\right) \\Lambda^\\pm(zp^{\\pm1}) G^+(w) \n- G^+(w) \\Lambda^\\pm(zp^{\\pm1}) f\\left( {z\\over w}\\right) \n&\\!\\!\\!=\\!\\!\\!& \\mp p^{\\mp1}(p^{3\\over 2}-p^{-{3\\over 2}})\n\\delta\\left( {w\\over z}p^{\\mp1}\\right) G^+(z) ,\\\\\nf\\left( {w\\over z}\\right) \\Lambda^\\pm(z) G^-(w) \n- G^-(w) \\Lambda^\\pm(z) f\\left( {z\\over w}\\right) \n&\\!\\!\\!=\\!\\!\\!& \\pm p^{\\pm2}(p^{3\\over 2}-p^{-{3\\over 2}})\n\\delta\\left( {w\\over z}p^{\\pm1}\\right) G^-(z) ,\n\\end{eqnarray*}\n\\vspace{-7mm}\n\\begin{eqnarray}\n\\left\\{G^+(z), G^-(w) \\right\\} \n&=&{p^{-3}\\over zw^2 (p-p^{-1})(p^{1\\over 2}-p^{-{1\\over 2}})} \\\\\n&\\times&\n\\left( \\Lambda^+\\left( {w}\\right)\\delta\\left( {w\\over zp}\\right) \n+ \\Lambda^-\\left( {w}\\right)\\delta\\left( {wp\\over z}\\right)\n-\\left( p^{1\\over 2}+p^{-{1\\over 2}}\\right)\n\\delta\\left( {w\\over z}\\right) \\right). \\nonumber\n\\end{eqnarray}\nTheir Fourier modes given by\n$G^\\pm(z) = \\sum_n G^\\pm_n z^{-n}$ and \n$\\Lambda^\\pm(z) = \\sum_n \\Lambda^\\pm_n z^{-n}$\nsatisfy \n$$\n\\left\\{G^+_{n+1}, G^-_{m+1} \\right\\} =\n{p^{-3}\\over (p-p^{-1})(p^{1\\over 2}-p^{-{1\\over 2}})}\n\\left( \\Lambda^+_{n+m-1} p^{-n} + \\Lambda^-_{n+m-1} p^n \n-\\left( p^{1\\over 2}+p^{-{1\\over 2}}\\right)\\delta_{n+m-1,0} \\right).\n$$\nNote that, \nthe relation between $G^+$ and $G^-$ can be \nexpressed in other ways.\nFor example, use the fact that \n$ G^+(z) G^-(w) {z\/w} + G^-(w) G^+(z) {w\/z} $ and\n$\\left\\{ G^+(z), G^-(w) \\right\\} z^{-r}{z^2 w^2\/(z+w)}$\nare also written by $\\Lambda^\\pm$ for any $r\\in{\\bf C}$.\n\nIn the $q=1$ case, $c=0$ topological algebra \ncan be constructed from \n$N=2$ superconformal algebra by the \noperation so-called ``twisting''\\cite{rEY}.\nWhat we have obtained here is a deformation of this twisted \nsuperconformal algebra. So far, \nwe have not been able to find a mechanism\nof ``untwisting'' in the deformed case.\nWe expect that our topological $q$-Virasoro algebra \nhelps us to find a \nsupersymmetric generalization of the $q$-Virasoro algebra \\v\nand a deformed twisting operation.\n\n\n\\subsection{$c=-2$ \\v and $c=1$ ${\\cal W}_{1+\\infty}$ algebra}\n\n\nSince $c=0$ Virasoro algebra is realized by the differential operator \n$L_n = - z^n D$ with $D = z \\partial_z$,\none can expect that $q$-Virasoro algebra \\v has a similar \nrepresentation by the difference \nor shift operator as\n$T_n \\sim z^n q^D$.\nHowever, this shift operator is nothing but the generating function of \nthe $c=0$ ${\\cal W}_{1+\\infty}$ generators.\nThus we expect some relations between the $q$-Virasoro and the ${\\cal W}_{1+\\infty}$ algebra.\nIndeed this is the case when $\\beta=2$, \nwe show a relation with $c=1$ ${\\cal W}_{1+\\infty}$ algebra.\n\nFirst, recall that the $q$-Virasoro generator is expressed by \nthe fundamental vertex operator and its dual $V_{2,1}^\\pm(z)$ \nas eq.\\ \\eq{eq:TSbyV21}.\nWhen $\\beta=2$, {\\it i.e.}, $c=-2$, ($q=1\/p$), \nthey reduce to the fermions such that\n\\begin{equation}\n\\{V_{2,1}^+ (z),V_{2,1}^- (w)\\}\n={1\\over z}\\delta\\left( {w\\over z}\\right),\\qquad\n\\{V_{2,1}^\\pm(z),V_{2,1}^\\pm(w)\\}=0.\n\\end{equation}\nOn the other hand, the generating function of $c=1$ ${\\cal W}_{1+\\infty}$ algebra\nis known to be also realized by a complex fermion.\nTherefore, we have found;\n\n\\proclaim Proposition 7.\nThe $\\beta=2$ ($c=-2$) $q$-Virasoro algebra $T(z) = \\Lambda^+(z) + \\Lambda^-(z)$ \ngenerates the $c=1$ ${\\cal W}_{1+\\infty}$ algebra\nand it is realized by fermions $V^\\pm_{2,1}(z)$ as follows;\n\\begin{equation}\n\\Lambda^\\pm(zq^{{1\\over 2}\\mp1}) =\n\\,:V_{2,1}^-(z) q^{\\mp D} V_{2,1}^+(z):\\, q^{\\mp{1\\over 2}}.\n\\label{eq:TbyFermion}\n\\end{equation}\n\n\nNext we show the relation between $q$-Virasoro and ${\\cal W}_{1+\\infty}$ algebras\nmore explicitly comparing their commutation relations.\nLet\n\\begin{equation}\nX^k(z) \\equiv\n\\,:\\exp\\left\\{\\sum_{n\\neq0}{1-q^{kn}\\over 1-q^n} h_n z^{-n} \\right\\}:\\, \nq^{k\\sqrt 2 h_0}\n\\equiv\n\\sum_{n\\in{\\bf Z}} X^k_n z^{-n}.\n\\end{equation}\nNote that the $q$-Virasoro generator is now $T(z)=X^1(z) + X^{-1}(z)$.\nThen\n\\begin{eqnarray}\\nonumber\n\\left[\\, X^k(z),X^\\ell(w) \\,\\right] \\&\n= {(q^k-1)(q^\\ell-1)\\over q^{k+\\ell}-1}\n\\left( X^{k+\\ell}(z) \\delta\\left( q^k{w\\over z}\\right) \n- X^{k+\\ell}(w) \\delta\\left( q^\\ell{z\\over w}\\right)\\right), \n\\cr\n\\left[\\, X^k(z),X^{-k}(w) \\,\\right] \\&\n= (q^{k\\over 2}-q^{-{k\\over 2}})^2 \\left( \n\\left( 1+\\sum_{n\\neq0}{1-q^{kn}\\over 1-q^n} \nn h_n z^{-n}\\right) \\delta\\left( q^k{w\\over z}\\right) \n-\\delta'\\left( q^k{w\\over z}\\right) \n\\right),\n\\end{eqnarray}\nfor $k,\\ell,n,m\\in{\\bf Z}$.\nTheir modes\n\\begin{equation}\nW^k_n = {X^k_n\\over q^k-1} - {1\\over 1-q^{-k}} \\delta_{n,0},\\qquad\nW^0_n = {n\\over q^n-1} h_n,\n\\end{equation}\nfor $k\\neq0$, satisfy\n\\begin{eqnarray}\n\\left[\\, W^k_n,W^\\ell_m \\,\\right] \\&\n= (q^{-km}-q^{-\\ell n}) W^{k+\\ell}_{n+m} + \n {q^{-km}-q^{-\\ell n}\\over 1-q^{-k-\\ell}} \\delta_{n+m,0},\n\\cr\n\\left[\\, W^k_n,W^{-k}_m \\,\\right] \\&\n= (q^{-km}-q^{kn}) W^0_{n+m} + n q^{kn}\\delta_{n+m,0}.\n\\label{eq:Winfty}\n\\end{eqnarray}\nThis is nothing but the algebra of \nthe generating functions for the $c=1$ ${\\cal W}_{1+\\infty}$ generators\n$W^k_n = W\\left( z^nq^{-kD}\\right)$\nin the notation of \\cite{rAFMO}.\n\nThe $c=0$ ${\\cal W}_{1+\\infty}$ algebra has a meaning of an area-preserving diffeomorphism\nand relates with classical membrane.\nWe expect this relation \nbetween the $q$-Virasoro algebra with the ${\\cal W}_{1+\\infty}$ algebra \nis a key for\na geometrical interpretation of \\v and\nthe quantization of the membrane.\n\n\n\n\n\\section{Summary and further issues}\n\n\nOur presentation has aimed to show that\nthe new Virasoro-type elliptic algebra \\v defined by eq.\\ \\eq{e:a1.2}\ncan be regarded as a universal symmetry of the massive integrable models.\n\n\nThe algebra \\v has two parameters $p$ and $q$ ($qp^{-1}=q^\\beta$)\nand it can be regarded as a generating function for \nseveral different Virasoro-type symmetry algebras\nappearing in solvable models.\nAt the limit of $q\\rightarrow 1$, the algebra \\v \nreduces to the ordinary Virasoro algebra \nwith the central charge $c$ related to $\\beta$ \\eq{e:a8}.\nWhen we consider the limit of $q\\rightarrow p$, \nit reduces to the \n$q$-Virasoro algebra of Frenkel-Reshetikhin \\eq{qvirFR}.\nWhen $q\\rightarrow 0$, \nwe obtain Jing's generating operators for\nthe Hall-Littlewood polynomials \\eq{eq:BosonHall}.\nThe topological algebra \\eq{eq:qTopological} and \nthe $c=1$ ${\\cal W}_{1+\\infty}$ algebra \\eq{eq:Winfty} are constructed from \nthe special cases $p=q^{-1\/2}$ and $p=1\/q$ respectively.\n\nOne of the peculiar features of the algebra \\v is its non-linearity.\nSince it is a quadratic algebra like the Yang-Baxter relation \nfor the transfer matrix,\nthe associativity is quite non-trivial \\eq{YB-for-T},\nand the Yang-Baxter equation \ndetermines the structure function uniquely, {\\it i.e.},\nfixes the algebra itself!\nMoreover it turns out to be the Zamolodchikov-Faddeev algebra\nof the particle-creation operators for the XYZ and \nthe sine-Gordon models \\eq{eq:ZFXYZ}.\n\nThe next essential nature is its infinite-dimensionality, \nwhich connects with the integrability of massive models.\nThe representation space of the algebra \\v possesses rich structure \nenough to describe the physical space of massive models.\nDespite its non-linearity, \nthe Kac determinant is very similar to that of the Virasoro case \n\\eq{Kacconj}. \n\n\nThe algebra \\v is realized by the free fields \n\\eq{eq:qVirFFR1} and \\eq{eq:qVirFFR2}\nin a quite simple way.\nIt shows us that the integrability of the model due to \\v symmetry\ncan be investigated in a natural manner in terms of the \nbosonic field.\nFurthermore this free field realization is described by \na deformed Miura transformation \\eq{eq:qMiura},\nand the deformed Miura transformation \nbrings about an interesting analogy with the dressed vacuum form.\nA generalization of this transformation gives\nthe $q$-deformed $\\cal W$ algebra \\cite{rFFr,rAKOS}.\n\nTo study infinite dimensional algebras, {\\it e.g.},\nnot only Virasoro and Kac-Moody algebras but also \\v,\none of the most essential oubjects in the representation theory is \nthe screening current. \nUsing the screening charge, one can\ndefine the physical states by the BRST method \\eq{eq:BRST},\nwrite down the null states \\eq{eq:qVirSing},\nand study the nontrivial monodromic property of the \nscreened vertex operators. \nThe null states \nrelate with the wave functions of the \nRuijsenaars model \\eq{eq:MacOp} which is \na relativistic generalization of the Calogero-Sutherland model. \nThe Hamiltonian of this model\nis realized by the positive modes of the \\v generators \\eq{e:b7.3}\nand causes the correspondence between the null states and excited states.\nThe monodromy matrix is connected to the ABF model.\nThe exchange relation of the vertex operators\npossesses a quantum group structure\ncharacterized by the solution of the face type \nYang-Baxter equation\\eq{eq:ExchangeABF}\nand leads to the identification of the deformed Virasoro vertex operators\nwith that of the ABF model.\nMoreover the screening currents show us an hidden symmetry;\nthat is an elliptic generalizaton of \n$k=1$ $U_q(\\widehat{sl}_2)$ algebra (Prop. 5).\n\\\\\n\n\nFinally, we mention some comments on further issues \ncoming from mathematical or physical points of view.\n\nMathematically, there are many things to be clarified.\nTo obtain a fusion of the ABF model or to find a suitable primary fields,\none needs a tensor product representation or \nsuitable adjoint action of the algebra \\v.\nIn other words, a co-product structure must be discovered. \nSince the algebra, however, is quadratic, it seems a highly non-trivial task.\nStudying the limit $q\\rightarrow 0$ may help us to reveal it.\n\nDispense with a help of free fields,\nthe correlation function must be determined\nby a difference equation coming from the Ward-Takahashi identity.\nTo find this equation, \nwe need to recognize a geometrical meaning of the algebra \\v;\nhow is a difference operator realized in \\v?\nRestricting the $q$-${\\cal W}_{1+\\infty}$ algebra \\cite{rKLR},\nwhich is defined as an algebra of higher pseud-difference operators,\nto the first order difference one,\nwe might obtain the algebra \\v and its difference operator realization.\nSeeking for a realization on the infinite $q$-wedge \\cite{rKMS}\nmight be able to connect both problems mentioned above; \na co-product and a geometrical interpretation.\n\nThe relation with the quantum affine Lie algebras also \nseems to be of some interest.\nFrenkel-Reshetikhin's $q$-Virasoro algebra was constructed \nby a $q$-Sugawara method \\cite{rFR} from \nthe $U_q(\\widehat{sl}_n)$ at the critical level.\nAre there any such constructions for the algebra \\v from \n$U_q(\\widehat{sl}_n)$ or, more hopefully, from \ntheir elliptic generalization ${\\cal A}_{q,p}$?\n\n\nPhysically, we anticipate many applications.\nThere are two approaches to investigate the 2-dimensional integrable models, \none is an Abelian method based on the algebraic Bethe-ansatz\nand the other is a non-Abelian one based on\nthe Virasoro algebra and its generalizations.\nThe latter approach describes the model more in detail,\nhowever its applicability is restricted \nonly to critical phenomena and some off-critical \ntrigonometric-type models.\nThe deformed Virasoro algebra \\v \nshould be a synthesis of the massive integrable models\nincluding elliptic-type models.\n\nIn the dual resonance model, which was a precursor of string theory,\nthe Veneziano amplitude has its generalizations\nto non-linearly rising Regge trajectories (see for example, \\cite{rFN}).\nHowever the absence of their operator representations\nhas disturbed their development, including to prove a no-ghost theorem. \nIn a special case, the amplitude reduces to $q$-beta function, \nwhich is very similar to a four-point function of our algebra \\v.\nWe hope that \\v gives \nan operator representation of a generalized Veneziano amplitude \nand opens further avenues to the exploration of new string theories.\n\nThe $c=0$ ${\\cal W}_{1+\\infty}$ algebra, an area-preserving diffeomorphism,\nis a symmetry of the classical membrane.\nThe relation between the $c=-2$ \\v algebra and $c=1$ ${\\cal W}_{1+\\infty}$ algebra \nmay be a key for\na geometrical interpretation of the algebra \\v and\nthe quantization of the membrane \nas the basic object of the $11(12)$-dimensional M(F) theory.\nOn the other hand, \nquantum membrane can be represented by a large $N$ matrix model \\cite{rDHN}, \nand the partition functions of more general conformal matrix models\nare described by eigenstates of the Calogero-Sutherland models \\cite{rAMOS}.\nDo the algebra \\v or eigenstates of the Ruijsenaars model \nrelate with a relativistic generalization of the quantum membrane? \n\nWhat is a field theoretical interpretation of the algebra \\v?\nThe low energy 4-dimensional $N=2$ super YM theories \nare described by some integrable models, \nperiodic Toda chain or elliptic CS model \\cite{rGKMMM}.\nIf we generalize them to 5D's one by Kaluza-Klein method\nwith an extra dimension compactified to a circle,\nthen they are described by the relativistic generalization of 4D's one\n\\cite{rN}.\nIt may suggest that the Kaluza-Klein with a radius $R$ can lead to \na relativistic generalization with the speed of light $1\/R$\nor a $q$-deformation with $q=e^R$ of a original theory.\nIs the algebra \\v understood as a Kaluza-Klein from CFT?\n\n\n\n\\vskip 5mm\n\n\\noindent{\\bf Acknowledgments:}\n\n\nWe would like to thank \nT.~Eguchi, D.~Fairlie, B.~Feigin, E.~Frenkel,\nJ.~Harvey, T.~Hayashi, T.~Inami, M.~Jimbo, \nS.~Kato, T.~Kawai, B.~Khesin, A.~Kuniba, \nE.~Martinec, Y.~Matsuo, T.~Miwa, K.~Nagatomo, N.~Nekrasov, \nP.~Wiegmann and Y.~Yamada\nfor discussions and encouragements.\nThis work is supported in part by Grant-in-Aid for Scientific\nResearch from Ministry of Science and Culture.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}