diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzmepk" "b/data_all_eng_slimpj/shuffled/split2/finalzzmepk" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzmepk" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nGiven a (finite, simple, undirected) graph $G=(V,E)$ we say that a subset of vertices $I \\subseteq V$ is \\emph{independent} if every two vertices in $I$ are non-adjacent. \nThe \\emph{maximum independent set problem} is the problem of finding an independent set of maximum cardinality in a given graph $G$. This problem is NP-hard even for planar graphs of degree at most three~\\cite{garey2002computers}, unit disk graphs~\\cite{DBLP:journals\/dm\/ClarkCJ90}, and $C_4$-free graphs~\\cite{alekseev1982effect}. ,\nTo see that the independent set problem is NP-hard in the class of $C_4$-free graphs, one can use the following observation by Poljak~\\cite{poljak1974note}. \nNamely, $\\alpha(G') = \\alpha(G) + 1$ where the graph $G'$ is obtained from $G$ by replacing a single\nedge with a $P_4$ (i.e., subdividing it twice). \nBy replacing every edge with a $P_4$ we obtain a graph that has girth at least nine, and thus MIS is NP-hard for $C_4$-free graphs.\nSimilarly, MIS is NP-hard for the class of graphs with girth at least $l$, where $l \\in \\mathbb{N}$ is fixed.\n\nOn the contrary, when the input is restricted to some particular class of graphs the problem can be solved efficiently. Examples of such classes are bipartite graphs~\\cite{DBLP:journals\/siamcomp\/HopcroftK73}, chordal graphs~\\cite{gavril1972algorithms} and claw-free graphs~\\cite{minty1980maximal,sbihi1980algorithme}. The maximum independent set problem is also polynomially solvable when the input is restricted to the class of perfect graphs using the ellipsoid method~\\cite{DBLP:journals\/combinatorica\/GrotschelLS81}, but it remains an open question to find a combinatorial algorithm\\footnote{The term combinatorial algorithm is used for an algorithm that does not rely on the ellipsoid method.} in this case. In fact, we do not even have a combinatorial FPT algorithm for the maximum independent set problem on perfect graphs.\n\n\\begin{sloppypar}\nClosely related to the class of perfect graphs is the class of even-hole-free graphs. The class of even-hole-free graphs was introduced as a class structurally similar to the class of Berge graphs. We say that a graph is \\emph{Berge} if and only if it is odd-hole-free and odd-antihole-free, i.e., $\\{C_5, C_7, \\overline{C_7}, C_9, \\overline{C_9}, \\dots\\}$-free\\footnote{Berge graphs are exactly perfect graphs by the Strong Perfect Graph Theorem.}. \nThe similarity follows from the fact that by forbidding $C_4$, we also forbid all antiholes on at least $6$ vertices. Hence, an even-hole-free graph does not contain an antihole on at least $6$ vertices, i.e., it is $\\{C_4, C_6, \\overline{C_6}, \\overline{C_7}, C_8, \\overline{C_8} \\dots\\}$-free.\nIt should be noted that techniques obtained in the study of even-hole-free graphs were successfully used in the proof of the Strong Perfect Graph Theorem. A decomposition theorem, an algorithm for the maximum weighted clique problem and several other polynomial algorithms for classical problems in subclasses of even-hole-free graphs can be found in survey~\\cite{vuvskovic2010even}.\n\\end{sloppypar}\n\nWe denote by $\\alpha (G)$ the maximum cardinality of an independent set in a graph $G$. In this paper we consider a parameterized version of the problem, that is we consider the following decision problem.\n\n\\begin{center}\n\\fbox{\\parbox{0.97\\linewidth}{\\noindent\n{\\sc Independent Set:}\\\\[.8ex]\n\\begin{tabular*}{.90\\textwidth}{rl}\n{\\bf Input:} & A graph $G$.\\\\\n{\\bf Parameter:}& $k$. \\\\\n{\\bf Output:} & {\\sc true} if $\\alpha(G) \\ge k$ and {\\sc false} otherwise.\n\\end{tabular*}\n}}\n\\end{center}\n\n\nWe say that a problem is \\emph{fixed parameter tractable} (FPT) parameterized by the solution size $k$, if there is an algorithm running in time $O(f(k)n^c)$ for some function $f$ and some constant $c$. \nMore generally, a problem is fixed parameter tractable with respect to the parameter $k$ (e.g. solution size, tree-width, ...) if for any\ninstance of size $n$, it can be solved in time $O(f(k)n^c)$ for some fixed $c$. \nUsually, we consider whether a problem is FPT if the problem is already known to be NP-hard. \nIn that case, the function $f$ is not in any way bounded by a polynomial. \nIn other words, for fixed parameter tractable problems, the difficulty is not in the input size, but rather in the size of the solution (parameter).\nIn general, the {\\sc Independent Set} problem \nis not fixed-parameter tractable (parameterized by the size of solution) unless W[1]$=$FPT or informally, we believe that there is no FPT algorithm for the problem~\\cite{downey2012parameterized}. \nRecently, it has been shown that MIS is W[1]-hard for $C_4$-free graphs~\\cite{bonnet_et_al:LIPIcs:2019:10218}. Even stronger, the same paper proves that MIS is W[1]-hard in any family of graphs defined by finitely many forbidden induced holes.\n\nWhile the exact complexity of the maximum independent set problem is still open for the class of even-hole-free graphs, we present a step forward by showing that there is an FPT algorithm for the problem.\n\n\\paragraph*{Main idea} Our algorithm is based on the augmentation technique. More precisely, in order to compute a solution of size $k+1$, we compute disjoint solutions of size $k$. The main property we use is that the union of two independent sets in an even-hole-free graph induces a forest. The key-point of our algorithm is that if $W,X$ are disjoint solutions of size $k$, and $Y$ is some (unknown) solution of size $k+1$, then the two trees induced by $X\\cup Y$ and $W\\cup Y$ are very constrained. This leads to a reduction to the chordal graph case, where MIS is tractable by dynamic programming.\n\n\\paragraph*{Preliminaries} \nWe consider finite, simple and undirected graphs. \nFor a graph $G=(V,E)$ we write $uv \\in E$ for an edge $\\{u,v\\} \\in E(G)$, in this case $u$ and $v$ are \\emph{adjacent}.\nFor a vertex $v \\in V(G)$ we denote by $N_G(v) = \\{u \\in V : uv \\in E\\}$ \\emph{the neighborhood} of $v$ and for $W \\subseteq V$, we define $N_G(W) = \\cup_{w \\in W} N_G(w) \\setminus W$. \nWe drop the subscript when it is clear from the context.\nLet $S \\subseteq V$. \nWe say that $S$ is \\emph{complete} to $W$ if every vertex in $S$ is adjacent to every vertex in $W$.\nThe \\emph{induced subgraph} $G[W]$ is defined as the graph $H = (W, E \\cap \\binom{W}{2})$ where $\\binom{W}{2}$ is the set of all unordered pairs in $W$. \nFor a set $A$ we denote by $A^2$ the set of all ordered pairs with elements in $A$.\nThe graph $G[V \\setminus W]$ is denoted $G\\setminus W$ and when $W=\\{w\\}$ we write $G \\setminus w$. \nA subset of vertices is called a \\emph{clique} if all the vertices are pairwise adjacent.\nA chordless cycle on at least four vertices is called a \\emph{hole}. A hole is even (resp. odd) if it contains an even (resp. odd) number of vertices.\nA \\emph{path} is a graph obtained by deleting one vertex of a chordless cycle. \nA path with endvertices $u, v$ is called a $u,v$-path.\nGiven a path $Z$ and two of its vertices $v,u$ we denote by $vZu$ the smallest subpath of $Z$ containing $v$ and $u$.\nAn \\emph{in-arborescence} is an orientation of a tree in which every vertex apart one (the \\emph{root}) has outdegree one.\n\n\\section{Reduction steps and augmenting graphs}\n\\label{section:MinimalAugmentingGraphs}\n\nOur main goal is to show that the following problem is FPT.\n\n\\begin{center}\n\\fbox{\\parbox{0.97\\linewidth}{\\noindent\n{\\sc Independent Set in Even-Hole-Free Graphs (ISEHF):}\\\\[.8ex]\n\\begin{tabular*}{.90\\textwidth}{rl}\n{\\bf Input:} & An even-hole-free graph $G$.\\\\\n{\\bf Parameter:}& $k$. \\\\\n{\\bf Output:} & An independent set of size $k$ if $\\alpha(G) \\ge k$ and {\\sc false} otherwise.\n\\end{tabular*}\n}}\n\\end{center}\n\nWe define a simpler version of the ISEHF problem where we know more about the structure \nof $G$. Later, we show that it suffices to find an FPT algorithm for the simpler version.\n\n\\begin{center}\n\\fbox{\\parbox{0.97\\linewidth}{\\noindent\n{\\sc Transversal Independent Set in Even-Hole-Free Graphs (TISEHF):}\\\\[.8ex]\n\\begin{tabular*}{.90\\textwidth}{rl}\n{\\bf Input:} & An even-hole-free graph $G$ and a partition of $V(G)$ into cliques $X_1,\\dots ,X_k$.\\\\\n{\\bf Parameter:}& $k$. \\\\\n{\\bf Output:} & An independent set of size $k$ if $\\alpha(G) \\ge k$ and {\\sc false} otherwise.\n\\end{tabular*}\n}}\n\\end{center}\n\nNote that in TISEHF, an independent set of size $k$ must intersect every clique on exactly one vertex, i.e., it must traverse all cliques.\n\n\\begin{lemma}\nThe ISEHF problem is FPT if and only if the TISEHF problem is FPT.\n\\end{lemma}\n\\begin{proof}\nNote that the only if implication is obvious, so we assume that we already have an FPT algorithm $\\cal A$ for TISEHF, and provide one for ISEHF. \nWe claim that it suffices to exhibit an algorithm $\\cal B$ running in time $g(k)n^c$ which takes as input the pair $(G,k)$ and either outputs an independent set of size $k$ or a cover of $V(G)$ by $2^{k-1}-1$ cliques. \nIndeed, one then just has to apply algorithm $\\cal A$ to every possible choice of $k$ disjoint cliques induced by the $2^{k-1}-1$ cliques which are output by $\\cal B$.\nWe describe $\\cal B$ inductively on $k$: If $k=2$, then $G$ is either a clique, or contains two non-adjacent vertices $x,y$. When $k>2$, we compute two non-adjacent vertices $x,y$ (or return the clique $G$). We now apply $\\cal B$ to the graph induced by the set $X$ of non-neighbors of $x$: we either get an independent set of size $k-1$ (in which case we are done by adding $x$) or cover $X$ by $2^{k-2}-1$ cliques. We apply similarly $\\cal B$ to the set $Y$ of non-neighbors of $y$. Note that $X\\cup Y$ covers all vertices of $G$ except the common neighbors $N$ of $x$ and $y$. Since $G$ is $C_4$-free, $N$ is a clique, and therefore we have constructed a cover of $V(G)$ by $2(2^{k-2}-1)+1$ cliques.\n\\end{proof}\n\nWe turn to our main result. \nIn the rest of this section we further reduce the problem to a graph together with two particular trees.\nSection~\\ref{section:bitrees} defines the notion of bi-trees and shows how two trees interact under certain conditions. \nThen, in Section~\\ref{section:end}, we prove that bi-trees arising from even-hole-free graphs satisfy these conditions and conclude the algorithm. \n\n\\begin{theorem}\\label{theo:trans}\nThe TISEHF problem is FPT.\n\\end{theorem}\n\n\\begin{proof}\nWe assume that we have already shown that there is an algorithm $\\cal A$ which solves TISEHF$(G,j)$ in time $O(f(j)n^3)$ for every $j\\leq k$. Our goal is to extend this by showing that $f(k+1)$ exists. Our input is a partition of $G$ into cliques $X_1,\\dots,X_k,X_{k+1}$ (which we call \\emph{parts}) and we aim to either find an independent set intersecting all parts or show that none exists. In what follows, we assume that an independent set $Y=\\{y_1,\\dots,y_k,y_{k+1}\\}$ intersecting all parts exists, and whenever a future argument will end up with a contradiction, this will always be a contradiction to the existence of $Y$, and thus our output will implicitly be {\\sc false}. \n\nThe first step is to apply $\\cal A$ to $X_1,\\dots,X_k$ to compute an independent set $W=\\{w_1,\\dots ,w_k\\}$. If it happens that $W\\cap Y\\neq \\emptyset$, we guess which $w_i$ belongs to $Y$ and run $\\cal A$ on the $k$ remaining parts in which we have deleted all neighbors of $w_i$. This costs $k$ calls to TISEHF$(G,k)$ which is in our budget. So we may assume that $W$ is disjoint from $Y$, and even stronger that no vertex of $W$ belongs to an independent set of size $k+1$, since one of the previous $k$ calls would have detected it. Moreover, since there is no even hole, $W\\cup Y$ induces a forest $T_1$. Note that no vertex of $W$ is isolated in $T_1$ since the parts are cliques. Note also that $T_1$ cannot have a leaf $w_i$ in $W$, since $w_i$ would belong to an independent set of size $k+1$ by exchanging it with $y_i$. Thus every vertex of $W$ has degree at least two in $T_1$. Since the number of edges of $T_1$ is at most $2k$, we have that every vertex of $W$ has degree 2 and $T_1$ is a tree.\n\nAs there is only $h(k)$ possible choices for the structure of $T_1$, we call $h(k)$ branches of computations for each of these choices of $T_1$. This means that in each call, we only keep the vertices of the parts $X_i$ which corresponds to the possible neighborhoods of vertices of $W$. For instance, in the call corresponding to a tree $T_1$ in which $w_1$ is adjacent to $y_1$ and $y_2$, we delete all neighbors of $w_1$ in parts $X_3,\\dots ,X_{k+1}$ and delete all non-neighbors of $w_1$ in $X_2$ (no further cleaning is needed in $X_1$ since it is a clique). Therefore, we assume that every vertex of $W$ is complete to exactly two parts (including its own) and non-adjacent to others. \nMoreover, we define a \\emph{white tree} on vertex set $\\{1,\\dots ,k+1\\}$ by having an edge between $i$ and $j$ if there exists a vertex $w$ of $W$ which is complete to $X_i$ and $X_j$. We will refer to this vertex $w$ as $w_{i,j}$. In what follows, we do not consider anymore that the vertices of $W$ belong to the parts $X_j$ and rather see them as external vertices of our problem. Thus, since we are free to rename the parts, we can assume that $k+1$ is a leaf of the white tree.\n\nThis is the crucial point of the algorithm, we have obtained a more structured input, but unfortunately we could not directly take advantage of it to conclude the main theorem. \nInstead, we apply again algorithm $\\cal A$ to $X_1,\\dots,X_k$ to compute a second independent set $X=\\{x_1,\\dots ,x_k\\}$ (if such an $X$ does not exist, we thus return {\\sc false} as $Y$ cannot exist). \nAs done previously, we may assume that $X$ is disjoint from $Y$, the tree $T_2$ spanned by $X\\cup Y$ can also be guessed, and the degrees of vertices of $X$ in $T_2$ is two (see Figure~\\ref{figure1}, down-left). \nWe now interpret $T_2$ in a slightly different way: we root $T_2$ at $y_{k+1}$ and orient all the edges toward the root. By doing so, every edge $\\{x_i,y_i\\}$ gives the arc $y_ix_i$ while the unique neighbor $y_{r(i)}$ of $x_i$, which is different from $y_i$, gives the arc $x_iy_{r(i)}$.\nWe now further clean the parts $X_j$ as follows: for every $x_i$, we delete all neighbors of $x_i$ in $X_j$ for $j\\neq i,r(i)$, and we delete all non-neighbors of $x_i$ in $X_{r(i)}$. \nWe now have two trees which endow our parts: the white tree and the \\emph{red in-arborescence} defined on vertex set $\\{1,\\dots ,k+1\\}$ by the arc set $\\{ir(i):i=1,\\dots, k\\}$. \nOur tool is now ready: the correlation between these two trees will provide an $O(k \\cdot n^3)$ time algorithm to compute $Y$, or show that $Y$ does not exist. We now turn to a special section devoted to bi-trees, i.e., trees defined on the same set of vertices under some structural constraints.\n\n\\section{Bi-trees}\n\\label{section:bitrees}\n\nLet $V$ be a set of vertices. A \\emph{bi-tree} is a triple $T = (V, A, E)$ where $E \\subseteq {V \\choose 2}$ is a set\nof edges such that $(V, E)$ is a tree and $A \\subseteq V^2$ is a set\nof arcs such that $(V, A)$ is an in-arborescence. For convenience, we view edges\nof $(V, E)$ as \\emph{white} edges, and arcs of $(V, A)$ as \\emph{red}\narcs. \n\nA \\emph{separation} of a bi-tree is a triple $(v, X, Y)$ such that:\n\\begin{itemize}\n\\item $V$ is partitioned into nonempty sets $\\{v\\}$, $X$ and $Y$,\n\\item no white edge has an end in $X$ and an end in $Y$, and\n\\item no red arc has an end in $X$ and an end in $Y$.\n\\end{itemize}\n\nWhen the sets $X$ and $Y$ are clear from the context, we will simply say that $v$ is a separation.\nNote that if $(v, X, Y)$ is a separation of a bi-tree $(V, E, A)$,\nthen $(X\\cup \\{v\\}, A \\cap (X\\cup \\{v\\})^2, E \\cap {X\\cup \\{v\\} \\choose 2})$ is the bi-tree \\emph{induced by $T\\setminus Y$}. Observe that if the root is not in $X$, then $T\\setminus Y$ is rooted at $v$.\n\nLet $T=(V, A, E)$ be a bi-tree and $a, b, v$ be three distinct\nvertices of $V$. Let $P_{ab}$ be a white path from $a$ to $b$, of\nlength one or two. Let $P_{av}$ be a directed red path, from $a$ to $v$,\nof length at least one. Let $P_{bv}$ be a directed red path, from $b$\nto $v$, of length at least one. We suppose that the three paths are\ninternally vertex disjoint (meaning that if a vertex is in at least\ntwo of the paths, then it must be $a$, $b$ or $v$). Three such paths\nare said to form an \\emph{obstruction directed to $v$}.\n\nLet $T=(V, A, E)$ be a bi-tree and $a, b, c, d$ be four distinct\nvertices of $V$. Let $P_{ab}$ be a white path from $a$ to $b$,\n$P_{bc}$ be a red path which is directed from $b$ to $c$ or from $c$ to\n$b$, $P_{cd}$ be a white path from $c$ to $d$ and $P_{da}$ be a red\npath which is directed from $d$ to $a$ or from $a$ to $d$. Suppose that\nat least one of $P_{ab}$, $P_{cd}$ has length exactly one and that\nthe four paths are internally vertex disjoint. Four such paths are\nsaid to form an \\emph{alternating obstruction}.\n\n A \\emph{bi-path} is a bi-tree $T = (V, A, E)$ on at least two vertices with an ordering $v_1, \\dots, v_n$ of $V$ and an integer $t$ such that:\n \\begin{itemize}\n \\item $A = \\{v_1v_2, \\dots, v_{n-1}v_n\\}$,\n \\item $v_1 v_n \\in E$,\n \\item $1 \\leq t \\leq n-1$,\n \\item if $t\\geq 2$, then $\\{v_1v_2, \\dots, v_1v_t\\}\\subseteq E$, and\n \\item if $t\\leq n-2$, then $\\{v_{t+1}v_n, \\dots, v_{n-1}v_n\\}\\subseteq E$.\n \\end{itemize}\n\n\n\\begin{lemma}\\label{lemma:isolatingPath}\n A bi-tree $T = (V, A, E)$ on at least two vertices, with no\n separation, no directed obstruction and no alternating obstruction is a bi-path.\n\\end{lemma}\n\\begin{proof}\n \\textit{Case~1}: $(V, A)$ contains some vertex with in-degree at\n least~2.\n\n We choose such a vertex $v$ as close as possible to the root $r$ of\n $(V, A)$. Since $(V, A)$ is an in-arborescence, $(V, A)\\setminus v$\n has at least $m\\geq 2$ in-components $A_1$, \\dots , $A_m$ and\n possibly one out-component $B$.\n By the choice of $v$, every vertex\n of $B$ has in-degree exactly 1. Therefore\n $(B \\cup \\{v\\}, A\\cap (B \\cup \\{v\\})^2)$ is a directed red path from $v$\n to $r$, that we call $Z$. We now state and prove two claims. \n\n \\begin{claim}\\label{claim1} \n For any $1 \\leq i < j \\leq m$, there is no white edge\n with one end in $A_i$ and one end in $A_j$. \n \\end{claim}\n \\begin{claimproof}\n Indeed, such an edge\n would yield an obstruction directed to $v$.\n \\end{claimproof}\n \\begin{claim}\\label{claim2}\n For every $1 \\leq i \\leq m$, there exists a white edge with\n one end in $A_i$ and one end in $B$ (so, in particular, $B$ exists).\n \\end{claim}\n \\begin{claimproof}\n For otherwise, Claim~\\ref{claim1} implies that\n $(v, A_i, V\\setminus (A_i \\cup \\{v\\})$ is a separation.\n \\end{claimproof}\n \n Let $P = v, \\dots ,z$ be the shortest white path such that $z\\in B$ where all internal vertices of $P$ are \n in $A_1\\cup \\dots \\cup A_m$ ($P$ has possibly length~1). \n By Claim~\\ref{claim1}, $P$ contains vertices from at most one component, say possibly $A_2$, among\n $A_1, \\dots, A_m$. By Claim~\\ref{claim2}, there exists a vertex $x\\in A_1$ with a white neighbor $w$ in $B$. Let $Q$ be the directed red path from $x$ to $v$.\n\n If $w$ is an internal vertex of $vZz$ then the edge $xw$, the\n directed path $wZz$, the path $P$, and the directed path $Q$ form an\n alternating obstruction. If $w$ is a vertex of $zZr$ different from\n $z$, then the edge $xw$, the directed path $zZw$, the path $P$, and\n the directed path $Q$ form an alternating obstruction. If follows\n that $w=z$.\n\n If $P$ has length greater than~1, then in particular $z$ has a white\n neighbor $y$ in $A_2$. Now, the white path $xzy$ and the\n in-components $A_1$ and $A_2$ yield an obstruction directed to $v$.\n So, $P$ has length~1. Consider, by Claim~\\ref{claim2}, a vertex $y'$ in $A_2$\n with a neighbor in $B$. The previous argument, with $A_1$ and $A_2$ interchanged, shows that $y'$ is adjacent to\n $z$ (just as we proved that $x$ is adjacent to $z$). Again, the\n white path $xzy'$ and the red in-components $A_1$ and $A_2$ yield an\n obstruction directed to $v$.\n\n \\medskip\n \\textit{Case~2}: Every vertex in $(V, A)$ has in-degree at most~1.\n\n Since $(V, A)$ is an in-arborescence, it follows that $(V, A)$ is a\n directed path. Hence, there exists an ordering $v_1, \\dots, v_n$ of\n the vertices of $T$ such that $A = \\{v_1v_2, \\dots, v_{n-1}v_n\\}$. \n\n \\begin{sloppypar}\n Suppose that there exists a white edge $v_iv_j$ with\n $1j$. It\n follows that $k=j$. We proved that there exists a white edge $v_{i'}v_j$,\n with $i'j$ and a white edge $v_iv_{j'}$. Now, the white edges\n $v_{i'}v_j$, $v_iv_{j'}$ and the red paths $v_{i'}\\dots v_{i}$ and\n $v_{j}\\dots v_{j'}$ form an alternating obstruction.\n \\end{sloppypar}\n\n Thus there is no white edge $v_iv_j$ with $11$ and $v_t v_n$\n is a white edge, then $(v_t, \\{v_1, \\dots,\n v_{t-1}\\}, \\{v_{t+1}, \\dots, v_n\\})$ is a separation. \n So, if $t>1$ then $v_1v_n$ is a white edge, and also if $t=1$. \n \\end{proof}\n\nGiven two bi-trees $T_1,T_2$ and a vertex $v$ of $T_1$, we denote by $(T_1,v, T_2)$ the bi-tree obtained by \\emph{gluing} $T_2$ at $v$ on $T_1$, i.e., by identifying the root of $T_2$ with $v$. \nA \\emph{bi-spider} is a bi-tree which is obtained by iteratively gluing bi-paths at the root vertex (see Figure~\\ref{figure1}, right; a bi-spider is induced by the set $\\{1,3,4,7,5\\}$).\nAlternatively, a bi-spider is a bi-tree with no directed obstruction and no alternating obstruction, which is either a bi-path or has only the root as a separation vertex. \n\nLet $T$ be a bi-tree with no directed obstruction and no alternating obstruction.\nNote that the previous lemma asserts that $T$ can be obtained by iteratively gluing bi-paths. \nIndeed, a separation $v$ which is chosen as far as possible from the root must isolate a bi-path. \n\nConsider a vertex $v$ of a bi-tree $T$. Since $T$ can be obtained by iteratively gluing bi-paths, if $v$ is not a separation then it is a vertex in $T$ which is not used in gluing. \nThus, the following property holds for $T$: every vertex $v$ which is not the root is either a separation vertex, a leaf of the white tree, or a leaf of the red in-arborescence. We use it to obtain the following result:\n \n\\begin{corollary}\\label{cor:main}\nA bi-tree $T = (V, A, E)$ on at least two vertices, with no directed obstruction and no alternating obstruction is either a bi-spider, or admits a separation $(v, X, Y)$ such that\n\\begin{enumerate}[(a)]\n\\item\\label{c1} $T\\setminus Y$ is a bi-spider,\n\\item\\label{c2} $v$ is either a leaf of the red in-arborescence induced by $T\\setminus X$ or a leaf of the white tree induced by $T\\setminus X$.\n\\end{enumerate}\n\\end{corollary}\n\\begin{proof}\nIf $T = (V, A, E)$ is not a bi-spider, it has a separation $(v, X, Y)$ distinct from the root, and we assume that among all choices, $v$ is chosen as far as possible from the root $r$ of the red in-arborescence. \nW.l.o.g., we assume that $Y$ contains $r$. \nThen $T\\setminus Y$ is a bi-tree rooted at $v$ which can only admit $v$ as a separation. \nHence, $T\\setminus Y$ is a bi-spider. \nAssume moreover that $Y$ is chosen minimum by inclusion for this property (equivalently, $T\\setminus Y$ is a maximum bi-spider rooted at $v$).\nWe claim that $v$ is not a separation in bi-tree $T\\setminus X$. \nIf $v$ is a separation in $T\\setminus X$ isolating a bi-path, then we have a contradiction to the minimality of $Y$. \nIf $v$ is a separation not isolating a bi-path, then we have a contradiction to the choice of $v$.\nHence, $T\\setminus X$ is a bi-tree in which $v$ is not a separation. \nSince $v$ is not the root either, it follows that $v$ is a white leaf or a red leaf in $T\\setminus X$. \n\\end{proof}\n\n\\begin{note}\\label{findBiSpider}\nA separation isolating a bi-spider with the properties~\\eqref{c1} and~\\eqref{c2} can be found efficiently. In particular, we find a separation $(v, X, Y)$ isolating a path and then take the maximal (inclusion-wise) set $X$ such that $T\\setminus Y$ is still a bi-spider.\n\\end{note}\n\n\\section{The end of the proof}\n\\label{section:end}\n\nWe now resume our proof of Theorem~\\ref{theo:trans} as follows. Lemma~\\ref{lemma:ehfNoObstructions} shows that the bi-trees arising from even-hole-free graphs do not have the obstructions. Hence, we can use the results from Section~\\ref{section:bitrees} where we proved that a bi-tree is either a bi-spider or has a separation isolating a bi-spider. Lemma~\\ref{lem:bi-spider} gives an algorithm for the problem when the underlying bi-tree is \na bi-spider. When the bi-tree is obtained by gluing bi-spiders, Lemma~\\ref{lem:bi-cut} proves that combining the partial solutions for each of the bi-spiders produces a valid solution.\n\nLet us recall the hypothesis of Theorem~\\ref{theo:trans} (see Figure~\\ref{figure1}):\n\n\\begin{enumerate}\n\\item The set of vertices of $G$ is partitioned into $k+1$ cliques $X_1,\\dots ,X_{k+1}$ and an additional set $W$ consisting of $k$ vertices $w_{a_1b_1},\\dots ,w_{a_kb_k}$.\n\\item Every $w_{a_ib_i}$ is completely joined to the two parts $X_{a_i}$ and $X_{b_i}$ and has no neighbor in the other parts.\n\\item The set of pairs $E=\\{\\{a_i,b_i\\}:i=1,\\dots ,k\\}$, seen as edges on the vertex set $V=\\{1,\\dots,k+1\\}$, forms a white tree in which $k+1$ is a leaf. \n\\item Every $X_i$, with $1\\leq i\\leq k$ contains a particular vertex $x_i$.\n\\item The set $\\{x_1,\\dots ,x_{k}\\}$ is an independent set.\n\\item For every vertex $x_i$, there is some $r(i)\\neq i$ such that $x_i$ is completely joined to $X_{r(i)}\\setminus x_{r(i)}$ (which is just $X_{r(i)}$ when $r(i)=k+1$).\n\\item The vertex $x_i$ is non-adjacent to every vertex of $X_j$, when $j\\neq i$ or $j\\neq r(i)$.\n\\item The set of ordered pairs $A=\\{(i,r(i)):i=1,\\dots ,k\\}$, seen as arcs on the vertex set $V=\\{1,\\dots ,k+1\\}$, forms a red in-arborescence rooted at $k+1$.\n\\end{enumerate}\n\nWe then have a bi-tree $T=(V,E,A)$ on the vertex set $V=\\{1,\\dots ,k+1\\}$. Furthermore, we want to decide if every part $X_i$, with $1\\leq i\\leq k+1$ contains a particular vertex $y_i$ distinct from $x_i$ and such that the set of these $y_i$'s forms an independent set.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width = 0.95\\textwidth]{figure1}\n\\caption{Up-left: Graph $G$. Down-left: Set of $y_i$'s. Up-right: White tree. Middle-right: Red in-arborescence. Down-right: Decomposition of bi-tree into bi-paths.}\n\\label{figure1}\n\\end{figure}\n\n\\begin{lemma}\\label{lemma:ehfNoObstructions}\nIf $G$ has no even holes and a set $Y$ exists, then $T=(V,E,A)$ has no directed obstruction and no alternating obstruction.\n\\end{lemma}\n\\begin{proof}\nLet us assume that we have a directed obstruction, i.e., we have three distinct vertices $a, b, v$ of $V$, a white path $P_{ab}$ from $a$ to $b$ of\nlength one or two, a directed red path $P_{av}$ of the form $a=a_0,a_1,\\dots ,a_r=v$, and a directed red path $P_{bv}$ of the form $b=b_0,b_1,\\dots ,b_s=v$. Our goal is to exhibit an even hole in $G$. The path $P_{ab}$ is either $ab$ or $acb$ and corresponds in $G$ to the path $P_1$ which is either $x_a,w_{ab},x_b$ or $x_a,w_{ac},y_c,w_{cb},x_b$. The path corresponding to $P_{av}$ is $P_2=x_{a_0},y_{a_1},x_{a_1},\\dots ,y_{a_r}$ and the path corresponding to $P_{bv}$ is $P_3=x_{b_0},y_{b_1},x_{b_1},\\dots ,y_{b_s}$. Note that $C=P_1\\cup P_2\\cup P_3$ is an even length cycle. \nMoreover, since each $x_i$ in $C$ is complete to only one class $X_j$ apart from its own, \nthere is no chord in $C$, a contradiction.\n\nLet us assume that we have an alternating obstruction on four distinct vertices $a, b, c, d$ of $V$. Two cases arise depending of the direction of the two red paths. When their directions are the same, we have a white path $P_{ab}$ from $a$ to $b$, a red path $P_{bc}$ directed from $b$ to $c$, a white path $P_{cd}$ from $c$ to $d$, and a red path $P_{ad}$ directed from $a$ to $d$. \nBy definition of alternating obstruction the four paths are internally vertex disjoint. \nAssuming that $P_{ab}$ is of the form $a=a_0,a_1,\\dots ,a_r=b$, we consider in $G$ the corresponding path $P_1=x_{a_0},w_{a_0a_1},y_{a_1},w_{a_1a_2},y_{a_2},w_{a_2a_3},\\dots ,x_{a_r}$. \nAssuming that $P_{bc}$ is of the form $b=b_0,b_1,\\dots ,b_s=c$, we consider in $G$ the corresponding path $P_2=x_{b_0},y_{b_1},x_{b_1},\\dots ,y_{b_s}$. \nAssuming that $P_{ad}$ is of the form $a=d_0,d_1,\\dots ,d_u=d$, we consider in $G$ the corresponding path $P_3=x_{d_0},y_{d_1},x_{d_1},\\dots ,y_{d_u}$. \nFinally, if $P_{cd}$ is of the form $c=c_0,c_1,\\dots ,c_v=d$, we consider in $G$ the corresponding path $P_4=y_{c_0},w_{c_0c_1},y_{c_1},w_{c_1c_2},y_{c_2},\\dots ,y_{c_v}$.\n\nWhen the red paths are in the opposite direction; we have a white path $P_{ab}$ from $a$ to $b$, a red path $P_{bc}$ directed from $b$ to $c$, a white path $P_{cd}$ from $c$ to $d$ and a red path $P_{da}$ directed from $d$ to $a$. \nAgain, the four paths are internally vertex disjoint. \nAssuming that $P_{ab}$ is of the form $a=a_0,a_1,\\dots ,a_r=b$, we consider in $G$ the corresponding path $P_1=y_{a_0},w_{a_0a_1},y_{a_1},w_{a_1a_2},y_{a_2},w_{a_2a_3},\\dots ,x_{a_r}$. \nAssuming that $P_{bc}$ is of the form $b=b_0,b_1,\\dots ,b_s=c$, we consider in $G$ the corresponding path $P_2=x_{b_0},y_{b_1},x_{b_1},\\dots ,y_{b_s}$. \nAssuming that $P_{da}$ is of the form $d=d_0,d_1,\\dots ,d_u=a$, we consider in $G$ the corresponding path $P_3=x_{d_0},y_{d_1},x_{d_1},\\dots ,y_{d_u}$. Finally, if $P_{cd}$ is of the form $c=c_0,c_1,\\dots ,c_v=d$, we consider in $G$ the corresponding path $P_4=y_{c_0},w_{c_0c_1},y_{c_1},w_{c_1c_2},y_{c_2},\\dots ,x_{c_v}$.\n\nNote that both $P_1,P_4$ are even length paths, and $P_2,P_3$ are odd length. Consequently $C=P_1\\cup P_2\\cup P_3\\cup P_4$ is an even length cycle. Moreover, no chord can arise so $C$ is an even hole, a contradiction.\n\\end{proof}\n\nBy Corollary~\\ref{cor:main}, the bi-tree $T=(V,E,A)$ is either a bi-spider, or has a separation $i$ isolating a bi-spider. We first conclude in the case of bi-spiders.\n\n\\begin{lemma}\n\\label{lem:bi-spider}\nIf $T$ is a bi-spider then there is an $O(n^3)$ time algorithm which computes $Y$ or shows that $Y$ does not exist.\n\\end{lemma}\n\n\\begin{proof}\nRecall that a bi-spider is a graph obtained by iteratively gluing bi-paths at the root vertex. \nDenote with $T_1, \\dots, T_l$ the bi-paths glued at the root vertex $k+1$ to obtain $T$. \nMoreover, assume that the in-arborescence $T_j$ is a directed path ${j}_1, \\dots, {j}_{s_j} = k+1$ for $1\\le j \\le l$.\nSince each $T_j$ is a bi-path, there is a vertex $w_{{j}_1, j_{s_j}}$ and for some value $t_j \\in \\{2,\\dots, s_j\\}$ (if any) we have the vertices $\\{w_{j_1,j_2},\\dots ,w_{j_1,j_{t_j}}\\}$ and $\\{w_{j_{t_j+1}, j_{s_j} }, \\dots ,w_{j_{s_j-1}, j_{s_j}}\\}$ (see Figure~\\ref{figure:bi-spider}). \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width = \\textwidth]{bi-spider}\n\\caption{An example for Lemma~\\ref{lem:bi-spider}.}\n\\label{figure:bi-spider}\n\\end{figure}\n\nWe decide if $Y$ exists in two phases.\nFirst, for every $1\\le j \\le l$ we find the set $Y_{j_1}$ of all vertices $y_{j_1}$ which are contained in an independent set of size $t_j$ intersecting $X_{j_1}, \\dots, X_{j_{t_j}}$. (Intuitively, $Y_{j_1}$ is the of vertices which can be extended to an independent set traversing $X_{j_1}, \\dots, X_{j_{t_j}}$, i.e., all the parts that have a common white neighbor with $y_{j_1}$ except $X_{k+1}$.)\nClearly, if $Y_{j_1}$ is empty for some $j$ then the set $Y$ does not exist. \n\nSecondly, we find the set $Y_{k+1}$ of vertices $y_{k+1}$ which are contained in an independent set of size $k- \\sum_{j=1}^{l} t_j$ intersecting $Y_{j_1}$ and $X_{j_{t_j + 1}}, \\dots X_{j_{s_{j}-1}}$ for all $1\\le j \\le l$. \n(Intuitively, $Y_{k+1}$ is the set of vertices which can be extended to an independent set traversing all the parts that have a common white neighbor with $y_{k+1}$.) Again, if $Y_{k+1}$ is empty then the set $Y$ does not exists. \n\nWe first assume that we have the sets $Y_{j_1}$'s and $Y_{k+1}$ and show how to conclude the lemma in this case. Later, we show that the sets are easy to find. \nLet $y_{k+1} \\in Y_{k+1}$ and let $\\displaystyle J = \\{y_{k+1}\\} \\cup_{j=1}^l \\{y_{j_1}\\} \\cup_{j=1}^l \\{y_{j_{t_j +1}}, \\dots, y_{j_{s_j -1}}\\}$ be an independent set of size $k- \\sum_{j=1}^{l} t_j$ intersecting all $Y_{j_1}$ and $X_{j_{t_j + 1}}, \\dots X_{j_{s_{j}-1}}$.\nFor each $1\\le j \\le l$, denote with $I_j = \\{y_{j_1}, \\dots, y_{j_{t_j}} \\}$ an independent set which contains $y_{j_1}$ and intersects $X_{j_1}, \\dots, X_{j_{t_j}}$.\nObserve that the set $Y = J \\cup_{j=1}^l I_j$ intersects each part of the graph. \nIt suffices to prove the following claim. \n\n\\begin{claim}\n $J \\cup_{j=1}^l I_j$ is also an independent set.\n\\end{claim}\n\\begin{claimproof}\nFor the sake of contradiction suppose otherwise. We consider two cases. \nEither there is an edge with one end in $J$ and the other end in $I_j$ for some $j$, or there is an edge with ends in $I_j$ and $I_i$ for some $1\\le i < j\\le l$. Let us deal with them respectively. \n\nThe mentioned edge is of the form $y_{j_p} y_{i_q}$ where $p \\le t_j$ and $t_i < q$ (possibly $j=i$) by definition of $I_j$ and $J$. Choose smallest such $p$.\nObserve that $p \\neq 1$ since $y_{j_1}$ is a vertex of both $J$ and $I_j$.\nIf $i_q \\neq k+1$ then\n$$y_{j_p}, x_{j_{p-1}}, \\dots, y_{j_2}, x_{j_1}, w_{j_1, j_{s_j}} (= w_{j_1, k+1}), y_{k+1}, w_{k+1, i_q} (= w_{i_{s_i}, i_q}), y_{i_q}$$ is a cycle of even length. \nMoreover, the cycle is induced by the choice of $p$ and since $\\{y_{j_2}, \\dots, y_{j_p} \\}$ is an independent set, a contradiction. An analogous situation arises if $i_q = k+1$.\n\nNow, we deal with the second case where there is an edge $y_{j_p} y_{i_q}$ where $p \\le t_j$, $q \\le t_i$ and $j \\neq i$. Choose largest such $q$.\nIt might happen that $p =1$ or $q=1$, but not both since $y_{j_1}, y_{i_1} \\in J$. Without loss of generality, $p \\neq 1$. \nThen $$y_{j_p}, x_{j_{p-1}}, \\dots, y_{j_2}, x_{j_1}, w_{j_1, j_{s_j}} (= w_{j_1, k+1}), y_{k+1}, x_{i_{s_i -1}}, y_{i_{s_i -1}}, \\dots, x_{i_q}, y_{i_q}$$ is an even cycle. \nBy the previous case there is no edge between $y_{k+1}$ and $I_j \\cup I_i$\nMoreover, by the choice of $q$, we deduce that the even cycle is induced, a contradiction.\n\\end{claimproof}\n\nIt remains to show how to find the sets $Y_{j_1}$'s and $Y_{k+1}$.\nFor the rest of the proof we only use the white tree. \nObserve that it suffices to prove the following (by setting $p = j_1$ for all $j$ and then $p=k+1$). \n\n\\begin{claim}\nLet $y_p \\in X_p$ and let $G'$ be the graph induced by $X_i$ such that $pi$ is an edge in the bi-tree $T$. Remove neighbors of $y_p$ in $G'$. Then $G'$ is chordal.\n\\end{claim} \n\\begin{claimproof}\nFor a contradiction, assume that $H$ is an odd hole in $G'$. \nEach part of $G'$ is a clique and, thus, contains at most two vertices of $H$. \nTherefore, there exist an induced path on three vertices $y_a,y_b,y_c$ of $H$, with $y_a, y_b, y_c$ in different parts $X_a, X_b, X_c$. By construction there are vertices $w_{p,a}, w_{p,b}$ and $w_{p,c}$. \nThen $y_{p}, w_{p, a}, y_a, y_b,y_c, w_{p, c}$ induces an even hole in $G$, a contradiction. \nSince $G$ is even-hole-free so is $G'$. Hence $G'$ is hole-free.\n\\end{claimproof}\nNow, for each $j$, we can check if $y_{j_1}$ is in $Y_{j_1}$ by finding a maximum independent set in $G' = G[\\cup_{i =2}^{t_j} X_i]\\setminus N(y_{j_1})$. The latter can be done in $O(n^2)$ since $G'$ is chordal~\\cite{gavril1972algorithms}. \nThen, we can check if $y_{k+1}$ is in $Y_{k+1}$ by finding a maximum independent set in $G' = G[\\cup_j \\{ Y_{j_1} \\cup_{i =t_j + 1}^{s_j-1} X_i \\}]\\setminus N(y_{k+1})$. This can be done in $O(n^2)$ since $G'$ is chordal.\nThe overall running time follows since each part is used exactly once in some $G'$.\n\\end{proof}\n\nIn fact, the previous algorithm gives a stronger result:\n\n\\begin{corollary}\n\\label{cor:cut} \nWhen $T$ is a bi-spider, there is an $O(n^3)$ time algorithm which computes all vertices $y_{k+1}$ which belong to an independent set of size $k+1$.\n\\end{corollary}\n\nWe now deal with the case when $i$ is a separation isolating a bi-spider. \nBy Corollary~\\ref{cor:main} bi-tree $T$ admits a separation $(i, B, C)$ isolating a bi-spider $T\\setminus C$ such that $i$ is either a red leaf or a white leaf in $T\\setminus B$.\nRecall that the vertex $k+1$ is a leaf of the white tree, hence, as a separation, $i$ is not equal to $k+1$.\nIn particular, the vertex $x_i$ exists. \nMoreover, since $T\\setminus C$ is a bi-spider it follows that $k+1 \\in C$. \nAs before, assuming the set $Y$ exists, we obtain the following lemma.\n\n\\begin{lemma}\\label{lem:bi-cut}\nThere is no edge from some $y_j$ with $j\\in B\\setminus i$ to some vertex $u \\in X_s$ with $s\\in C$. \n\\end{lemma}\n\\begin{proof}\nWe denote by $r$ the root of $T$. As argued above $r \\in C$ ($r= k+1$).\nFor the sake of contradiction suppose that there is an an edge $y_ju$.\n\nLet us consider bi-spider $T\\setminus C$.\nThere is a red path $j=j_0,\\dots ,j_a=i$ in $(V, A)$ which can be turned into an induced path $P_0=y_{j_0},x_{j_0},y_{j_1},x_{j_1},\\dots ,y_{j_a},x_{j_a}$ in $G$ from $y_j$ to $x_i$ with odd length. \nThere is also a white path $j=b_0,\\dots ,b_d=i$ in $(V, E)$ which can be turned into an induced path $P_1=y_{b_0},w_{b_0b_1},y_{b_1},w_{b_1b_2},\\dots ,x_{b_d}$ in $G$ from $y_j$ to $x_i$ with even length. \nNow, in order to conclude the lemma it suffices to find a $u, x_i$ path $P$ such that $P.P_0$ and $P.P_1$ induce cycles. Then, since $P_0$ and $P_1$ are of different parity a contradiction arises. \nIn the rest of the proof we show how to find $P$.\n \nFirst, observe that since $T\\setminus C$ contains a white subtree, $u$ is non-adjacent to $y_i$ or to any $y_q$ where $q\\in B$ and $q\\neq j$ since it would yield an even hole (there is an even path between any two different vertices $y_p, y_q$). Hence, $u$ is adjacent to $y_j$ and non-adjacent to all other vertices in $P_0$ and $P_1$.\n\nBy Corollary~\\ref{cor:main}, $i$ is either a red leaf or a white leaf in $T\\setminus B$. We consider two cases.\n\n\\smallskip\n\\textit{Case 1:} $i$ is a red leaf. Then there is a (an undirected) red path $i = i_0, \\dots, i_s = s$ in $T\\setminus B$, which can be turned into an induced path $P = x_{i_0}, y_{i_1}, x_{i_1}, \\dots, x_{i_{s-1}, u}$ in $G$ from $x_i$ to $u$. By construction, this path is induced. Moreover, since $i$ is a red leaf in $T\\setminus B$ it follows that $y_{i_1} \\neq y_i$. Therefore, both $P.P_0$ and $P.P_1$ induce cycles, i.e., there is no chord with one end in $P$ and the other in $P_0$ or $P_1$.\n\n\\smallskip\nNote that the same argument holds whenever the red path $i = i_0, \\dots, i_s = s$ does not contain $y_i$.\nHence, the only remaining case is when $i$ is on the red directed path from $s$ to $r$ in $T\\setminus B$.\nDenote with $Q$ the directed red $sr$ path in $T\\setminus B$. \n\n\\smallskip\n\\textit{Case 2:} $i$ is a white leaf and $i \\in Q$. \nLet $Q' = iQr$ be subpath of $Q$ starting at $i$ and ending at $r$. \nSince $i$ is not a separation of $T\\setminus B$, there exists a white path $s = s_0, \\dots, s_t$ \nconnecting $s$ and $Q'$. Moreover, the path does not contain $i$. \nWe choose the shortest such $s,Q'$ path. \nThis path can be turned into an induced path $P_2 = u, w_{s_0, s_1}, y_{s_1}, \\dots, w_{s_{t-1}, s_t}, y_{s_t}$ in $G$ with endpoints $u$ and $y_{s_t}$.\nBy the above $y_i \\not \\in P_2$ and also no vertex $w_{i, . }$ is used in $P_2$.\n\nConsider the directed path $iQ s_t$ $(= i Q' s_t)$. Denote it as $i=i_0, i_1, \\dots, i_{l} = s_t$. It can be turned into an induced path $P_3 = x_{i_0}, y_{i_1}, \\dots y_{i_l}$ in $G$ with endpoints $x_i$ and $y_{s_t}$. \nThen $P_2. P_3$ is a $u, x_i$ path in $G$. The concatenation $P_2. P_3$ might not be an induced path, but we can shorten it to obtain an induced $u x_i$ path $P$ in $G$. Now, it can be checked that $P.P_0$ and $P. P_1$ induce cycles since $P$ does not use $y_i$ or any of the vertices $w_{i, .}$. \n\\end{proof}\n\nWe are now ready to show that there is an $O(k \\cdot n^3)$ time algorithm which computes $Y$ when $T=(V,E,A)$ is a bi-tree. If $T$ is a bi-spider, we are done by Lemma~\\ref{lem:bi-spider}. Otherwise, by Corollary~\\ref{cor:main}, there is a separation $(i, B, C)$ which isolates a bi-spider $T\\setminus C$.\nBy Lemma~\\ref{lem:bi-cut}, one can delete all vertices $y_j \\in X_j$ for $j\\in B\\setminus i$ with a neighbor $u \\in X_k$ for $k\\notin B$, and this reduction is sound since no candidate $y_j$ can have such an edge. Now, by Corollary~\\ref{cor:cut}, one can compute in $O(n^3)$ time the set $X'_i \\subseteq X_i$ of vertices, each of which extends, in the bi-spider $T\\setminus C$, to an independent set of size $|B|$. \nFrom the bi-spider $T\\setminus C$, we only keep these vertices $X'_i$. \nObserve that the number of parts has now decreased by at least one. \nWe repeat this process until we either construct $X'_{k+1}$ or conclude that this set is empty. \nIf $X'_{k+1} \\neq \\emptyset$, then we can reconstruct the set $Y$.\nThe total time is $O(k \\cdot n^3)$.\n\\end{proof}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{\\label{sec:contributions}Author's Contributions}\nA.R., S.W. and I.T. designed the project. A.R. and P.B. performed the experiments and the simulations. All authors contributed to the analysis of the results and to the writing of the manuscript.\n\n\n\\section*{\\label{sec:contributions}Acknowledgements}\nI.T. and P.K.B. acknowledge financial support from the Swiss National Science Foundation (SNF) through the grant No. 200021-179312.\nA. R. is much obliged to Vladimir Nikolaevitch Smirnov who financially supported his work.\nAll authors would like to acknowledge support from the IBM Q network and thank the qiskit development team for discussions regarding the development of the software.\n\n\nIBM, IBM Q, Qiskit are trademarks of International Business Machines Corporation, registered in many jurisdictions worldwide. Other product or service names may be trademarks or service marks of IBM or other companies.\n\n\n\\section*{\\label{sec:M&M}Materials and Methods}\n\nThe noisy simulations were conducted with $\\alpha=1\\%$ (resp. $\\alpha=0.1\\%$) for the $128$ shots (resp. $1024$ shots) simulation with an ``all-to-all`` entangling scheme on Qiskit~\\cite{Qiskit}. \nAll circuits were constructed with a VQE depth of $m=2$. Given a run with $n$ qubits, the size of the population for the evolutionary algorithm was set to $P=5mn$, a typical size according to the literature. The selection strategy of the DE algorithm is practically identical to the original ``current-to-best\/1\/bin'' \\cite{das_recent_2016}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction and Notations}\nLet $(A,\\mathfrak m)$ be an excellent two-dimensional normal local ring \nand let $I$ be an $\\mathfrak m$-primary ideal of $A$. \nThe integral closure $\\bar{I}$ of $I$ is the ideal consisting of all solutions $z$ of some equation with coefficients $c_i \\in I^i $: \n$Z ^n +c_1 Z^{n-1} +c_2 Z^{n-2} + \\dots+ c_{n-1} Z + c_n=0$. \nThen $ I \\subseteq \\bar{I} \\subseteq \\sqrt{I}$. \nWe say that $I$ is \\textit{integrally closed} if $I = \\bar{I}$ and $I$ is \\textit{normal} if $I^n= {\\overline{I^n}} $ for every positive integer $n$. \nBy a classical result of Rees \\cite{R}, under our assumptions, \nthe filtration $\\{\\overline{I^n}\\}_{n \\in \\mathbb N}$ is a good $I$-filtration of $A$ \nand it is called the \\textit{normal filtration}. \n\\par \\vspace{1mm}\nWe may define the Hilbert Samuel function $\\bar{H}_I(n):=\\ell_A(A\/\\overline{I^{n+1}})$ for all integers $n \\ge 0$ and it becomes a polynomial for large $n$ \n(here $\\ell_A(M) $ is the length of the $A$-module $M$). \nThis polynomial is called the \\textit{normal Hilbert polynomial}\n\\[\n\\bar{P}_I(n) =\\bar e_0(I) \\binom{n+2}{2}\n-\\bar e_1(I) \\binom{n+1}{1}\n+\\bar e_2(I), \n\\]\nand the coefficients $\\overline{e}_i(I)$, $i=0,1,2,$ are the \n\\textit{normal Hilbert coefficients}. \n\n\\par \nA rich literature is available on the normal Hilbert coefficients $\\bar e_i(I)$ and this study is considered an important part of the theory of blowing-up rings, see for instance \n\\cite{CPR1, CPR2, Hun, It1, It2, MOR, MORT, MSV, RV}. \n\\par \nFrom the geometric side, any integrally closed $\\mathfrak m$-primary \nideal $I$ of $A $ is represented on some resolution, see \\cite{Li}. \nLet $$f \\colon X \\to \\Spec A$$ be a resolution of singularities \nwith an anti-nef cycle $Z>0$ on $X$ so that \n$I =I_Z= H^0(\\mathcal{O}_X(-Z))$ \n and $I\\mathcal{O}_X=\\mathcal{O}_X(-Z)$. \nWe say that $I = I_Z$ is \\textit{represented by $Z $ on $X$}. \nThe aim of this paper is to join the algebraic \nand the geometric information on $A$ \ntaking advantage of the theory of the Hilbert functions and \nof the theory of the resolution of singularities. \n\\par \nFor a coherent $\\mathcal{O}_X$-module $\\mathcal{F}$, we write \n$h^i(\\mathcal{F})=\\ell_A(H^i(X, \\mathcal{F}))$. \nIf $I=I_Z$ is an $\\mathfrak m$-primary integrally closed ideal of $A$ \nrepresented by $Z$ on $X$, one can define for every integer \n$n \\ge 0$ a decreasing chain of integers \n$q(nI):= q(\\overline{I^n}) = h^1(\\mathcal{O}_X(-nZ))$ \nwhere $q(0I) := p_g(A)$ is the \\textit{geometric genus} of $A$. \nIt is proved that $q(nI) $ stabilises for every $I$ and $n\\ge p_g(A)$. \nWe denote it by $q(\\infty I)$. \n\\par \nThese integers are independent of the representation \nand they are strictly related to the normal Hilbert polynomial. \nThe key of our approach can be considered Theorem \\ref{kato} and \nProposition \\ref{p:normalHP}, consequences of Kato's Riemann-Roch \nformula (see \\cite{kato} and \\cite{OWY2}). \nIn particular the following holds:\n\\begin{enumerate}\n\\item $\\overline{P}_I(n)=\\ell_A(A\/\\overline{I^{n+1}})$ for all $n \\ge p_g(A)-1$. \n\\item \n$\\bar e_1(I)-e_0(I) + \\ell_A(A\/I) =p_g(A) - q(I)$. \n\\item $\\bar e_2(I)= p_g(A)-q(nI)=p_g(A)-q(\\infty I)$ for all $n \\ge p_g(A)$. \n\\end{enumerate}\nMoreover, we have \n\\[\n\\bar e_0(I)= -Z^2,\\qquad \\bar e_1(I)= \\dfrac{-Z^2+ZK_X}{2}. \n\\]\n\\par \nThis makes the bridge between the theory of the normal Hilbert \ncoefficients and the theory of the singularities. \nThis is the line already traced by Lipman \\cite{Li}, Cutkosky \\cite{C} \nand more recently by Okuma, Watanabe and Yoshida, \nsee \\cite{OWY1, OWY2, OWY3}. \n\\par \nLet $(A,\\mathfrak m)$ be a two-dimensional excellent normal local domain \ncontaining an algebraically closed field $k= A\/\\mathfrak m$. \nIt is known that $A$ is a rational singularity (see \\cite{Ar}) if and only if \nevery integrally closed $\\mathfrak m$-primary ideal $I$ of $A$ is normal \n(see \\cite{Li} and \\cite{C}), equivalently $\\bar e_2(I)=0$, that is \n$I$ is a $p_g$-ideal, as proved in \\cite{OWY1, OWY2}. \nInspired by a paper by the first author \\cite{Ok}, we investigate the \nintegrally closed $\\mathfrak m$-primary ideals of \\textit{elliptic singularities} \n(see Wagreich \\cite{W}) and of \\textit{strongly elliptic singularities} \n(see Yau \\cite{Y}). \nAll the preliminary results are contained in Section 2. \n\n\\vskip 1mm\nIn Section 3 we prove the main results of the paper. \nWe define {\\it{the elliptic and the strongly elliptic ideals}} \naimed by the study of non rational singularities. \nWe recall that if $Q$ is a minimal reduction of $I$, then \nwe denote by \n$\\br(I):= \\min\\{ r \\;|\\; \\overline{I^{n+1}} = Q \\overline{I^n}\\;\n\\text{for all} \\; n \\ge r\\}$ \\textit{the normal reduction number} of $I$ \nand this integer exists and does not depend on the choice of $Q$. \nOkuma proved that if $A$ is an elliptic singularity, then \n$\\br(I) = 2$ for any integrally closed $\\mathfrak m$-primary ideal of $A$, \nsee \\cite[Theorem 3.3]{Ok}. \nAccording to Okuma's result, we define \\textit{elliptic ideals} \nto be the integrally closed $\\mathfrak m$-primary ideals satisfying $\\br(I)= 2$. \nIn Theorem \\ref{nr2} we prove that elliptic ideals satisfy \n$\\bar e_2(I)=\\bar{e}_1(I) -e_0(I) + \\ell_A( A\/I)>0 $ attaining the minimal \nvalue according to the inequality proved by Sally \\cite{S2} and Itoh \n\\cite{It2}. \nIn particular if $I$ is an elliptic ideal, then $p_g(A) > q(I)=q(\\infty I)$. \nIf $A$ is not a rational singularity, then elliptic ideals always exist, see Proposition \\ref{exist}. In particular we prove\n\\vskip 1mm\n\\noindent {\\bf{Proposition.}} (See Proposition 3.3.) \n{\\it{If $A$ is not a rational singularity, then \nfor any $\\mathfrak m$-primary integrally closed ideal $I$ of $A$, \n$\\overline{I^{n}}$ is either a $p_g$-ideal or \nan elliptic ideal for every $n \\ge p_g(A)$. }}\n\n\\par \\vspace{2mm}\nYau in \\cite{Y}, Laufer in \\cite{La} and Wagreich in \\cite{W} \nintroduced interesting classes of elliptic singularities. \nAn excellent two-dimensional normal local ring \n$A$ is a {{strongly elliptic singularity}} if $ p_g(A)=1$, \nthat is $p_g$ is almost minimal. \n\\par \nAmong the elliptic ideals, in Theorem \\ref{e_2=1} we define strongly elliptic ideals those for which $\\bar e_2 =1 $ and equivalent conditions are given. \nThe following result characterizes algebraically the strongly elliptic singularities. \n\\vskip 1mm\n\\noindent {\\bf{Theorem.}} (See Theorem 3.14.) \n\\textit{Let $(A,\\mathfrak m)$ be a two-dimensional excellent normal local domain containing an algebraically closed field $k= A\/\\mathfrak m$ \n and assume that $p_g(A)>0$. \nThe following conditions are equivalent:}\n\\begin{enumerate}\n\\item \\textit{$A$ is a strongly elliptic singularity.} \n\\item \\textit{Every integrally closed ideal of $A$ is either a $p_g$-ideal or a strongly elliptic ideal.}\n\\end{enumerate}\n\\par \nNotice that $p_g$-ideals are always normal, but \nelliptic ideals are not necessary normal, see Proposition \\ref{normal}, Examples \\ref{exnormal} and \\ref{noti}. \nMoreover if $A$ is strongly elliptic and $I$ is not a $p_g$-ideal, then \nProposition \\ref{normal} and Theorem \\ref{I2} give necessary and sufficient \nconditions for being $I$ normal.\n\\vskip 1mm\n\\noindent {\\bf{Theorem.}} {\\it{ Let $(A,\\mathfrak m)$ be a two-dimensional \nexcellent normal local domain \ncontaining an algebraically closed field $k= A\/\\mathfrak m$. \nAssume that $A$ is a strongly elliptic singularity. \nIf $I = I_Z $ is an elliptic ideal \n$($equivalently $I$ is not a $p_g$-ideal$)$ and $D$ is \nthe minimally elliptic cycle on $X$, then $I^2$ is integrally closed \n$($equivalently $I$ is normal$)$ if and only if $- Z D \\ge 3$ and \nif $- ZD \\le 2$, \nthen $I^2 = QI$. \n}}\n\\vskip 1mm\nFor any normal surface singularity which is not rational, $p_g$-ideals and \nelliptic ideals exist plentifully. \nBut this is no longer true for strongly elliptic ideals. \n\\par \\vskip 1mm\nIn Section 4, we show that there exist excellent two-dimensional normal \nlocal rings having no strongly elliptic ideals, see Examples \\ref{no}. \nFinally, Corollary \\ref{existence} gives necessary and sufficient conditions \nfor the existence of strongly elliptic ideals in terms of the existence of \ncertain cohomological cycles. \nWhen there exist, we present an effective geometric construction, see \nExample \\ref{si}. \n\n\\section{Preliminaries and normal reduction number}\n \nLet $(A,\\mathfrak m)$ be an excellent two-dimensional normal local domain \ncontaining an algebraically closed field $k= A\/\\mathfrak m$ and let $I$ be an \nintegrally closed $\\mathfrak m$-primary ideal of $A$. \nWith the already introduced notation, then there exists a resolution \n$X \\to \\ \\Spec A$ and a cycle $Z$ such that $I$ is represented on $X$ by \n$Z$. \nWhen we write $I_Z$ we always assume that $\\mathcal{O}_X(-Z) $ is generated by \nglobal sections, namely $I \\mathcal{O}_X=\\mathcal{O}_X(-Z)$, and note that \n$I_Z= H^0(X, \\mathcal{O}_X(-Z))$. \nRecall that the geometric genus $p_g(A)= h^1(\\mathcal{O}_X)$ is \nindependent of the choice of the resolution. \n\\par \nOkuma, Watanabe and Yoshida introduced a natural extension of the \nintegrally closed ideals in a two-dimensional rational singularity, \nthat is the $p_g$-ideals. With the previous notation\n\\[\np_g(A) \\ge h^1(\\mathcal{O}_X(-Z) )\n\\]\nand if the equality holds, then $Z$ is called a $p_g$-cycle and $I=I_Z$ is \ncalled a $p_g$-ideal. \nIn \\cite{OWY1, OWY2}, the authors \ncharacterized the $p_g$-ideals in terms of the normal Hilbert polynomial. \nThey proved that $A$ is a rational singularity if and only if every integrally \nclosed $\\mathfrak m$-primary ideal is a $p_g$-ideal. \nStarting by $p_g(A)$ we define the following chain of integers. \n\n\\begin{defn} \\label{qI} \nWe define $q(I):= h^1(\\mathcal{O}_X(-Z))$ and more in general \n$q(nI):= q(\\overline{I^n}) = h^1(\\mathcal{O}_X(-nZ))$ for every \ninteger $n \\ge 1$. \n\\end{defn}\n\\par \nWe put $q(0 I)=h^1(\\mathcal{O}_X)=p_g(A)$. \nNotice that $q(nI)$ is in general very difficult to compute, but it is independent of the representation \\cite[Lemma 3.4]{OWY1}. \nThese invariants are strictly related to the normal Hilbert polynomial and their interplay will be very important in our approach.\n\\par \nThe following formula is called a Riemann-Roch formula. \nThe result was proved in \\cite{kato} in the complex case, \nbut it holds in any characteristic, see \\cite{WY}. \n\n\\begin{thm}[\\textbf{Kato's Riemann-Roch formula \\cite[Theorem 2.2]{WY}}] \n\\label{kato}\nLet $I=I_Z$ be an $\\mathfrak m$-primary integrally closed ideal represented \nby an anti-nef cycle $Z$ on $X$. \nThen we have \n\\[\n\\ell_A(A\/I) + q(I) =-\\dfrac{Z^2+K_XZ}{2} + p_g(A),\n\\]\nwhere $K_X$ denotes the canonical divisor. \n\\end{thm}\n\n\\par\nWe recall here the properties of the sequence $\\{q(nI)\\}$, \nThe following Propositions \\ref{q(nI)} and \\ref{eq:q(nI)} \non $I = I_Z$ follows from the long exact sequence \nattached to the short exact sequence \n\\[\n(\\dagger) \\quad 0\\to \\mathcal{O}_X(-(n-1)Z) \\to \\mathcal{O}_X(-nZ)^{\\oplus 2}\\to \\mathcal{O}_X(-(n+1)Z)\\to 0.\n\\]\nsee \\cite[Lemma 3.1]{OWY2}. \n\n\\begin{prop} \\label{q(nI)}\nWith the previous notation, the following facts hold: \n\\begin{enumerate}\n\\item $0 \\le q(I) \\le p_g(A);$ and \n\\item $q(kI) \\ge q((k+1)I)$ for every integer $k \\ge 0$ and \nif $q(nI) = q((n+1)I)$ for some $n\\ge 0$, \nthen $q(nI) = q(mI)$ for every $m\\ge n$. \nHence $q(nI) = q((n+1)I)$ for every $I$ and $n\\ge p_g(A)$. \nWe denote it by $q(\\infty I)$. \n\\end{enumerate}\n\\end{prop}\n\n\\par \nWe will use the above sequence for computing the following important \nalgebraic numerical invariants of the normal filtration \n$\\{\\overline{I^n}\\}$. \nLet $\\mathbb{Z}_{+}$ denote the set of positive integers. \n\n\\begin{defn}[\\textrm{cf. \\cite{OWY4}}] \\label{nrns}\nLet $I \\subset A$ be $\\mathfrak m$-primary integrally closed ideal, \nand let $Q$ be a minimal reduction of $I$. \nDefine:\n\\begin{eqnarray*}\n\\nr(I) &:=& \\min\\{r \\in \\mathbb{Z}_{+} \\,|\\, \\overline{I^{r+1}}=Q\\overline{I^r}\\}, \\\\\n\\br(I) &:=& \\min\\{ r \\in \\mathbb{Z}_{+} \\;|\\; \\overline{I^{n+1}} = Q \\overline{I^n} \\; \\mbox{\\rm for all } n \\ge r\\}.\n\\end{eqnarray*}\nWe call $\\br(I)$ the {\\it{normal reduction number}} and \n$\\nr(I)$ the \\textit{relative normal reduction number}. \n\\end{defn}\n\n\\par \nThe normal reduction number exists (see \\cite{NR} and \\cite{R}) and it \nhas been studied by many authors in the context of the Hilbert function \nand of the Hilbert polynomial (e.g. see \\cite{Hun}, \\cite{It2}, \\cite{CPR1}, \n\\cite{CPR2}, \\cite{MSV}). \nThe main difficulty of the normal filtration with respect to the $I$-adic \nfiltration, is that the Rees algebra of the normal filtration is not generated \nby the part of degree one because $I \\overline{I^n} \\neq \n\\overline{I^{n+1}}$. By the definition, we deduce that $\\nr(I) \\le \\br(I)$ \nand we will see that in general they do not coincide. \nNote that the definitions of $\\nr(I)$ and of $\\br(I)$ are independent on the \nchoice of a minimal reduction $Q$ of $I$ \n(see e.g. \\cite[Theorem 4.5]{Hun}). \nIt is also a consequence of the following result in \\cite{OWY4}, \\S2. \n\n\n\\begin{prop}\\label{eq:q(nI)}\nThe following statements hold. \n\\begin{enumerate}\n\\item \nFor any integer $n \\ge 1$, we have \n\\[\n2 \\cdot q(nI) + \\ell_A(\\overline{I^{n+1}}\/Q\\overline{I^n})\n=q((n+1)I)+q((n-1)I). \n\\]\n \\item We have\n\\begin{eqnarray*}\n\\nr(I) &=& \\min\\{n \\in \\mathbb{Z}_{+} \\,| \\, \n q((n-1)I)- q(nI) = q(nI) - q((n+1)I) \\},\\\\\n\\br(I) &=& \\min\\{n \\in \\mathbb{Z}_{+} \\,|\\, q((n-1)I)=q(nI) \\}.\n\\end{eqnarray*}\n\\end{enumerate}\n\\end{prop}\n\n\\par \nFrom the propositions above we have that $\\br(I) \\le p_g(A) +1$.\nIn \\cite[Theorem 2.9]{OWY4} the authors showed that \n$p_g(A) \\ge \\binom{\\nr(I)}{2}$. \n\n\\par \nThe second author \\cite[Corollary 1.5]{Ro} proved the following upper \nbound on the reduction number $r(I)$ for every $\\mathfrak m$-primary ideal $I$ \n(here $r(I)$ denotes the reduction number for the $I$-adic filtration) in a \ntwo-dimensional Cohen-Macaulay local ring $A$ in terms of the Hilbert \ncoefficients: \n\\[\nr(I) \\le e_1(I) -e_0(I) + \\ell_A(A\/I)+1. \n\\] \nThe bound gives, as a consequence, several interesting results, \nin particular a positive answer to a longstanding conjecture stated \nby J. Sally in the case of local Cohen-Macaulay rings of \nalmost minimal multiplicity, see \\cite{Ro} and \\cite{S1}. \nLater, the inequality was extended by Rossi and Valla, see \\cite{RV}, \nTheorem $4.3$ to special multiplicative $I$-filtrations. \nThe result does not include the normal filtration. \nIt is natural to ask if the same bound also holds for $\\br(I)$. \nThe answer is negative as we will show later, but we prove that the \nanalogue upper bound holds true for $\\nr(I)$. \nWe need some preliminary results. \n\n\\par \\vspace{1mm}\nFrom Riemann-Roch formula (Theorem $\\ref{kato}$), we get \n\\[\n\\ell_A(A\/\\overline{I^{n+1}})+q((n+1)I)\n=-\\dfrac{(n+1)^2Z^2+(n+1)ZK_X}{2} + p_g(A). \n\\]\nUsing this, \nwe can express $\\bar{e}_0(I), \\bar{e}_1(I), \\bar{e}_2(I)$ as follows. \n\\begin{prop}[\\textrm{\\cite[Theorem 3.2]{OWY2}}] \\label{p:normalHP}\nAssume that $I = I_Z$ is represented by a cycle $Z > 0$\n on a resolution $X$ of $\\Spec(A)$. \nLet $\\bar{P}_I(n)$ be the normal Hilbert-polynomial of $I$. \nThen \n\\begin{enumerate}\n\\item $\\overline{P}_I(n)=\\ell_A(A\/\\overline{I^{n+1}})$ for all \n$n \\ge p_g(A)-1$. \n\\item $\\bar e_0(I)=e_0(I)$.\n\\item $\\bar e_1(I)-e_0(I) + \\ell_A(A\/I) =p_g(A) - q(I)$. \n\\item $\\bar e_2(I)= p_g(A)-q(nI)=p_g(A)-q(\\infty I)$ for all $n \\ge p_g(A)$. \n\\end{enumerate}\nMoreover, we have \n\\[\n\\bar e_0(I)= -Z^2,\\qquad \\bar e_1(I)= \\dfrac{-Z^2+ZK_X}{2}. \n\\]\n\\end{prop}\n\n\\begin{thm} \\label{nr(I)}\nLet $(A,\\mathfrak m)$ be an excellent two-dimensional \nnormal local domain containing an algebraically closed field $k= A\/\\mathfrak m$. \nLet $I \\subset A$ be an $\\mathfrak m$-primary integrally closed ideal. \nThen \n\\[\n\\nr(I) \\le \\bar{e}_1(I) - \\bar{e}_0(I) +\\ell_A(A\/I) +1. \n\\]\nIf we put $r=\\nr(I)$, \nequality holds if and only if \nthe following conditions hold true $:$\n\\begin{enumerate}\n\\item $\\ell_A(\\overline{I^{n+1}}\/Q\\overline{I^n}) =1$\nfor $n= 1,\\ldots, r-1$ if $r>1,$ \n\\item $q((r-1)I)=q(\\infty I)$.\n\\end{enumerate}\nWhen this is the case, $\\nr(I)=\\br(I)$, $q(I)=p_g(A)-\\br(I)+1$, \nand $\\bar{e}_2(I)=p_g(A)- q(\\infty I) = r(r-1)\/2$. \n\\end{thm}\n\n\\begin{proof} \nBy virtue of Proposition \\ref{p:normalHP}, \nit is enough to show \n\\[\n\\nr(I) \\le p_g(A)-q(I)+1. \n\\]\nIf we put $\\Delta q(n) : = q(nI) - q((n+1)I)$ for every integer $n \\ge 0$, \nthen $\\Delta q(n) $ is non-negative and decreasing \nsince $\\ell_A(\\overline{I^{n+1}}\/Q\\overline{I^n})\n=\\Delta q(n-1) -\\Delta q(n)$.\nWe have \n\\[\n\\nr(I)=\\min\\{n \\in \\mathbb{Z}_{+} \\,|\\, \\Delta q(n-1) =\\Delta q(n) \\}, \n\\quad\n\\br(I)=\\min\\{n \\in \\mathbb{Z}_{+} \\,|\\, \\Delta q(n-1) =0 \\}. \n\\]\nPut $a=p_g(A)-q(I)$. \nThen $\\Delta q(0)=a\\ge \\nr(I)-1$ and $\\nr(I)=a+1$ if and only if \n\\[\n\\Delta q(0) =a > \\Delta q(1) =a-1 > \\cdots > \\Delta q(a-1)=1 > \n\\Delta q(a)=0 = \\Delta q(a+1). \n\\]\n\\par \nNow assume $\\nr(I)=a+1$. Then $a=r-1$ and for every $n$ with \n$1 \\le n \\le a=r-1$, we have \n\\[\n\\ell_A(\\overline{I^{n+1}}\/Q\\overline{I^n})=\\Delta q(n-1)-\\Delta q(n)=\n(a-n+1)-(a-n)=1. \n\\]\nMoreover, for every $n \\ge a+1$, we have \n\\[\n\\ell_A(\\overline{I^{n+1}}\/Q\\overline{I^n})=\\Delta q(n-1)-\\Delta q(n)=0\n\\]\nand thus $\\overline{I^{n+1}}=Q\\overline{I^n}$. \nHence $\\br(I)=a+1=\\nr(I)$. \nFurthermore, \n\\[\n\\bar{e}_2(I)=p_g(A)-q(\\infty I)=q(0 I)-q((r-1)I) = \\sum_{i=0}^{r-2} \\Delta q(i)= \n\\dfrac{r(r-1)}{2}. \n\\] \nOne can prove the converse similarly. \\qed\n\\end{proof}\n\n\\par \\vspace{1mm}\nNote that, if the equality holds in the previous result, then the normal \nfiltration $\\{\\overline{I^n}\\} $ has almost minimal multiplicity \nfollowing the definition given in \\cite[2.1]{RV}. \nIn the following example we show that \nTheorem \\ref{nr(I)} does \\textit{not} hold if we replace $\\nr(I)$ \nby $\\bar{r}(I)$. \nThe example shows that for all $g \\ge 2$, \nthere exist an excellent two-dimensional normal local ring $A$ \nand an integrally closed $\\mathfrak m$-primary ideal $I$ \nsuch that $\\nr(I) =1, \\bar{r}(I)={{g+1}}, q(I) = g-1$ and \n$\\ell_A(A\/I) = g$. \n\n\\par\nThe following ideal $I$ satisfies $\\bar{e}_1(I)=\\bar{e}_0(I)-\\ell_A(A\/I)+1$, \nbut $\\br(I)\\not \\le 2$. \n\n\\begin{ex}[\\textrm{\\cite[Example 3.10]{OWY5}}] \\label{e}\nLet $g \\ge 2$ be an integer, and let $K$ be \n a field of $\\chara K=0$ or $\\chara K=p$, where \n $p$ does not divide $2g+2$. \nThen $R=K[X,Y,Z]\/(X^2-Y^{2g+2}-Z^{2g+2})$ is \na graded normal $K$-algebra with $\\deg X=g+1$, $\\deg Y=\\deg Z=1$. \nLet $A=R^{(g)}$ be the $g^{th}$ Veronese subring of $R$:\n\\[\nA=K[y^{g},y^{g-1}z,y^{g-2}z^{2},\\ldots, z^{g}, xy^{g-1},xy^{g-2}z,\n\\ldots,xz^{g-1}],\n\\]\nwhere $x, y, z$ denotes, respectively, the image of $X, Y, Z$ in $R$. \nThen $A$ is a graded normal domain with $A_k=R_{kg}$ \nfor every integer $k \\ge 0$. \nLet $I=(y^g,y^{g-1}z) +A_{\\ge 2}$ and $Q=(y^g-z^{2g}, y^{g-1}z)$. \nThen the following statements hold: \n\\begin{enumerate}\n\\item $p_g(A)=g$. \n\\item $\\nr(I)=1$ and $\\br(I)=g+1$. Indeed, \n\\begin{enumerate}\n\\item $\\overline{I}=I$ and $\\overline{I^{n}}=I^n=QI^{n-1}$ \nfor every $n=2,\\ldots, g$. \n\\item $\\ell_A(\\overline{I^{g+1}}\/Q\\overline{I^g})=1$\n($\\overline{I^{g+1}}= I^{g+1} +(xy^{g^2-1})$).\n\\item $\\overline{I^{n+1}}=Q \\overline{I^{n}}$ for every $n \\ge g+1$. \n\\end{enumerate}\n\\item $\\bar{e}_0(I)=4g-2$, $\\bar{e}_1(I)=3g-1$, $\\bar{e}_2(I)=g$ and \n$\\ell_A(A\/I)=g$. \n\\item $q(nI)=g-n$ for every $n=0,1,\\ldots,g$; $q(gI)=q(\\infty I)=0$. \n\\end{enumerate}\n\\end{ex}\n\nThe first statement follows from that $a(A)=0$ and $g=g(\\Proj (A))$.\nFor the convenience of the readers, \nwe give a sketch of the proof in the case of $g=2$ \n(see \\cite[the proof of Example 3.10]{OWY5}). \nLet $A=K[y^2,yz,z^2,xy,xz]=R^{(2)}$ with $\\deg x=3$ and \n$\\deg y= \\deg z=1$, and \n$I=(y^2,yz, z^4, xy,xz) \\supset Q=(y^2-z^4,yz)$. \nThen one can easily see that $e_0(I)=\\ell_A(A\/Q)=4g-2=6$, \n$\\ell_A(A\/I)=p_g(A)=g=2$ and $I^2=QI$, $\\overline{I}=I$. \nIn particular, $\\nr(I)=1$. \n\n\\begin{clm}\n$f_0 \\in K[y,z]_{2n} \\cap \\overline{I^n} \n\\Longrightarrow f_0 \\in I^n$ for each $n \\ge 1$. \n\\end{clm}\n\n\\par \nThe normality of $I_0=(y^2,yz,z^4)K[y,z] \\subset K[y,z]$ implies the above \nclaim. \n\n\n\\begin{clm}\n$0 \\ne f_1 \\in K[y,z]_{2n-3}$, \n$xf_1 \\in \\overline{I^n} \\Longrightarrow n \\ge 3$. \n\\end{clm}\n\n\\par \nBy assumption and Claim 1, \nwe have $(y^6+z^6)f_1^2=(xf_1)^2 \\in \\overline{I^{2n}} \n\\cap K[y,z]_{2 \\cdot 2n} \\subset I^{2n}$. \nThe degree (in $y$ and $z$) of any monomial in \n$I^{2n}=(y^2, yz, z^4, xy, xz)$ is at least $4n=\\deg (y^6+z^6) f_1^2$. \nHence $(y^6+z^6)f_1^2 \\in (y^2, yz)^{2n}$ and \nthe the highest power of $z$ appearing in $(y^6+z^6) f_1^2$ is \nat most $2n$. Therefore $n \\ge 3$. \n\n\\begin{clm}\nIf $n \\le 2$, then $\\overline{I^n} \\cap A_n \\subset I^n \\cap A_n$. \n\\end{clm}\n\n\\par \nAny $f \\in \\overline{I^n} \\cap A_n$ can be written as \n$f=f_0+xf_1$ for some $f_0 \\in K[y,z]_{2n}$ and $f_1 \\in K[y,z]_{2n-3})$. \nLet $\\sigma \\in \\Aut_{K[y,z]^{(2)}}(A)$ \nsuch that $\\sigma(x)=-x$. \nThen since $\\sigma(I)=I$, we obtain $\\sigma(f)=f_0 - xf_1 \\in \n\\overline{I^n}$. Hence \n\\[\nf_0 = \\frac{f+\\sigma(f)}{2} \\in \\overline{I^n} \n\\quad \\text{and} \\quad \nxf_1 =\\frac{f-\\sigma(f)}{2} \\in \\overline{I^n}.\n\\] \nBy Claims 1,2, we have $f_0 \\in I^n$ and $f_1=0$. \nTherefore $f=f_0 \\in I^n \\cap A_n$, as required. \n\n\\begin{clm}\n$xy^3 \\in \\overline{I^3} \\setminus Q\\overline{I^2}$. \n\\end{clm}\n\nSince $(xy^3)^2=(y^6)^2+(y^3z^3)^2 \\in (I^3)^2$, \nwe get $xy^3 \\in \\overline{I^3}$. \nAssume $xy^3 \\in Q\\overline{I^2}=(a,b)\\overline{I^2}$, \nwhere $a=y^2-z^4$ and $b=yz$. \nThen $axy+bxz^3=xy^3=au+bv$ for some $u,v \\in \\overline{I^2}$. \nSince $a,b$ forms a regular sequence, we can take an element \n$h \\in A_1$ so that $u-xy=bh$ and $xz^3-v =ah$. So we may \nassume $u,v \\in A_2$, and thus \n$u,v \\in \\overline{I^2} \\cap A_2 \\subset I^2$. \nHowever, this yields $xy^3=au+bv \\in QI^2=I^3$, which is a \ncontradiction. \n\n\n\\begin{clm}\n$q(I)=1$, $q(2I)=q(\\infty I)=0$, \n$\\ell_A(\\overline{I^3}\/Q\\overline{I^2})=1$ and \n$\\overline{I^{n+1}}=Q\\overline{I^n}$ for each $n \\ge 3$. \n\\end{clm}\n\nBy Proposition \\ref{q(nI)}, we have \n$2=p_g(A)=q(0\\cdot I)\\ge q(I) \\ge q(2\\cdot I) \\ge 0$. \nIf $q(I)=q(2\\cdot I)$, then $q(2 \\cdot I)=q(3 \\cdot I)$. \nThis implies $\\ell_A(\\overline{I^3}\/Q\\overline{I^2})=0$ \nfrom Proposition \\ref{eq:q(nI)}. This contradicts Claim 4. \nHence $q(I)=1$ and $q(2\\cdot I)=0$. \nThe other assertions follow from Proposition \\ref{eq:q(nI)}. \nIn particular, $\\br(I)=3$. \n\n\n\\begin{clm}\n$\\bar e_1 (I) =3g-2=5$, $\\bar e_2(I)=g=2$. \n\\end{clm}\n\nBy Proposition \\ref{p:normalHP}, we have \n\\begin{eqnarray*}\n\\bar e_1(I)&=& e_0(I)-\\ell_A(A\/I)+p_g(A)-q(I)=6-2+2-1=5, \\\\\n\\bar e_2(I) &=& p_g(A)-q(\\infty I)=~2-0=2. \n\\end{eqnarray*}\n\\section{Elliptic and Strongly Elliptic ideals}\n\nWe define the Rees algebra $\\bar{\\mathcal R}(I)$ and the \nassociated graded ring $\\bar{G}(I)$ associated to the normal filtration \nas follows:\n\n\\begin{eqnarray*}\n\\bar{\\mathcal R}(I) \n&:= & \\bigoplus_{n\\ge 0}\\overline{I^n} t^n \\subset A[t]. \\\\\n\\bar{G}(I) &:= & \\bigoplus_{n\\ge 0}\\overline{I^n}\/ \\overline{I^{n+1}} \\cong \n\\bar{\\mathcal R}(I)\/\\bar{\\mathcal R}(I)(1). \n\\end{eqnarray*}\n\\par \\vspace{2mm}\n$\\bar{\\mathcal R}(I)$ (resp. $\\bar{G}(I)$) is called the \n\\textit{normal Rees algebra}\n(resp. \\textit{the normal associated graded ring}) of $I$. \nWe recall that the $a$-invariant of a graded $d$-dimensional ring \n$R$ with maximal homogeneous graded ideal $\\mathfrak{M}$ \nwas introduced by \\cite{GW} and \ndefined as $a(R):= \\max\\{n | [H^d_{\\mathfrak M}(R)]_n \\neq 0\\}$, \nwhere $[H^d_{\\mathfrak M}(R) ]_n$ denotes the homogeneous component of \ndegree $n$ of the graded $R$-module $H^d_{\\mathfrak M}(R)$. \n\\par \nIt is known that $A$ is a rational singularity if and only if \n$\\overline r(A)=1$, see \\cite[Proposition 1.1]{OWY5}. \nIn \\cite{OWY1,OWY2}, the authors introduced the notion of $p_g$-ideals, \ncharacterizing rational singularities. \n\n\\begin{thm}[\\textrm{cf. \\cite{OWY1, OWY2, GN, Hun}}] \\label{r=1}\nLet $(A,\\mathfrak m)$ be a two-dimensional excellent normal non-regular \nlocal domain containing an algebraically closed field $k= A\/\\mathfrak m. $\nLet $I=I_Z$ be an $\\mathfrak m$-primary integrally closed ideal of $A$. \nPut $\\bar{G}=\\bar{G}(I)$ and $\\bar{\\mathcal R}=\\bar{\\mathcal R}(I)$. \nThen the following conditions are equivalent$:$\n\\begin{enumerate}\n\\item $\\br(I)= 1$. \n\\item $q(I)=p_g(A)$. \n\\item $I^2=QI$ and $\\overline{I^n}=I^n$ for every $n \\ge 1$. \n\\item $\\bar e_1(I) = e_0(I)-\\ell_A(A\/I)$. \n\\item $\\bar e_2(I)=0$. \n\\item $\\bar{G}$ is Cohen-Macaulay with $a(\\bar{G})< 0$. \n\\item $\\bar{\\mathcal R}$ is Cohen-Macaulay. \n\\end{enumerate}\nWhen this is the case, $I$ is said to be a \\textit{$p_g$-ideal}. \n\\end{thm}\n\n\\begin{proof} \nSince $QI^{n-1} \\subset I^n \\subset \\overline{I^n}$ \nfor every $n \\ge 2$, (1) $\\Leftrightarrow (3)$ is trivial. \n(1) $\\Leftrightarrow$ (5) (resp. (6) $\\Leftrightarrow$ (7)) follows from \n\\cite[Part II, Proposition 8.1]{GN} \n(resp. \\cite[Part II, Corollary 1.2]{GN}).\nMoreover. the equivalence of (4),(5) and (7) follows from \n\\cite[Part II, Theorem 8.2]{GN}. \n(2) $\\Leftrightarrow$ (4) follows from Proposition $\\ref{eq:q(nI)}$. \\qed\n\\end{proof}\n\n\\par \nIt is known that $A$ is a rational singularity if and only if \nany integrally closed $\\mathfrak m$-primary ideal is a $p_g$-ideal\n (see \\cite{OWY1, OWY2}). \nWe define \n\\[\n\\overline{r}(A):=\\max\\{\\br(I) \\,|\\, \\text{$I$ is an integrally closed \n$\\mathfrak m$-primary ideal}\\}. \n\\]\nThen $A$ is a rational singularity if and only if \n$\\overline r(A)=1$, see \\cite[Proposition 1.1]{OWY5}. \n \n\\par \\vspace{1mm}\nOkuma proved in \\cite[Theorem 3.3]{Ok} \nthat if $A$ is an elliptic singularity, then $\\overline r(A)=2$.\nFor the definition of elliptic singularity we refer to \\cite[page 428]{W} \nor \\cite[Definition 2.1]{Ok}. \nWe investigate the integrally closed $\\mathfrak m$-primary ideals such that \n$\\br(I) =2$ with the aim to characterize elliptic singularities. \nNext result extends and completes a result by S. Itoh \n\\cite[Proposition 10]{It2}, by using a different approach. \n\n\\begin{thm} \\label{nr2}\nLet $(A,\\mathfrak m)$ be a two-dimensional excellent normal local domain \ncontaining an algebraically closed field $k= A\/\\mathfrak m$ and let \n$I \\subset A$ be an $\\mathfrak m$-primary integrally closed ideal. \nPut $\\bar{G}=\\bar{G}(I)$ and $\\bar{\\mathcal R}=\\bar{\\mathcal R}(I)$. \nThen the following conditions are equivalent$:$\n\\begin{enumerate}\n \\item $\\br(I)=2$. \n \\item $p_g(A) > q(I)=q(\\infty I)$. \n \\item $\\bar{e}_1(I) = e_0(I) -\\ell_A( A\/I) + \\bar{e}_2(I)$ \n and $\\bar{e}_2(I) >0$.\n \\item $\\ell_A(A\/\\overline{I^{n+1}}) = \\bar{P}_I(n)$ for all $n\\ge 0$ and $\\bar{e}_2(I) >0$.\n \\item $\\bar{G}$ is Cohen-Macaulay with $a(\\bar{G})=0$.\n\\end{enumerate}\nWhen this is the case, $I$ is said to be an \\textbf{elliptic ideal} and \n $\\ell_A ([H^2_{\\mathfrak M}(\\bar{G})_0) = \\ell_A(\\overline{I^2}\/QI) = \\bar{e}_2(I)$. \n \\end{thm} \n \n\\begin{proof} \n$(1) \\Longleftrightarrow (2):$ It follows from Proposition\n$\\ref{eq:q(nI)}$(2). \n\n\\par \\vspace{2mm} \n\\noindent $(2) \\Longleftrightarrow (3):$ \nBy Proposition \\ref{p:normalHP} we have\n\\begin{eqnarray*}\n\\bar{e}_1(I) &=& e_0(I) -\\ell_A( A\/I) + \\bar{e}_2(I) \n- \\bigg\\{q(I)-q(\\infty I)\\bigg\\}. \\\\\n\\bar{e}_2(I) &=& p_g(A) - q(\\infty I) \\ge 0. \n\\end{eqnarray*}\nThe assertion follows from here. \n\n\\par \\vspace{2mm} \n$(2) \\Longleftrightarrow (4):$ \nAssume $I= I_Z = H^0(X,\\mathcal{O}_X(-Z))$ \nfor some resolution $X \\to \\Spec A$. \nBy Kato's Riemann-Roch formula, for every integer $n \\ge 0$, we have\n\\[\n\\ell_A(A\/\\overline{I^{n+1}})+h^1(\\mathcal{O}_X(-(n+1)Z))\n=-\\dfrac{(n+1)^2Z^2+(n+1) K_XZ}{2} +p_g(A). \n\\]\nHence \n\\begin{eqnarray*}\n\\ell_A(A\/\\overline{I^{n+1}}) &=& \\bar{e}_0(I){n+2 \\choose 2} \n- \\bar{e}_1(I) {n+1 \\choose 1} + \\bigg\\{p_g(A)-q((n+1)I) \\bigg\\} \\\\\n&=& \\bar{P}_I(n)- \\bigg\\{q((n+1)I)-q(\\infty I) \\bigg\\}.\n\\end{eqnarray*}\n\n\\par \nAssume (4). \nBy replacing $0$ to $n$ in the above equation, \nwe get $q(I)=q(\\infty I)$, hence (2). \nConversely, if $q(I)=q(\\infty I)$, then since $q((n+1)I)=q(\\infty I)$ \nfor all $n \\ge 1$, the above equation implies (4). \n\n\\par \\vspace{2mm} \n\\par \\noindent $(1) \\Longrightarrow (5):$ \nPut $Q=(a,b)$. \nSince $\\overline{I^{n+1}} \\colon a=\\overline{I^n}$, \n$a^{*}$, the image of $a$ in $\\bar{G}$ is a non zero divisor of $\\bar{G}$.\n\\par \nBy assumption, we have \n$\\overline{I^{n+1} } \\cap Q = Q {\\overline{I^{n}}} \\cap Q\n=Q {\\overline{I^{n}}}$ for every $n \\ge 2$. \nOn the other hand, we have \n$\\overline{I^2} \\cap Q = QI$ \nby \\cite[Theorem in page 371]{Hun} or \\cite[Theorem]{It1}. \nThen it is well-known that\n$a^{*}$, $b^{*}$ forms a regular sequence in $\\bar{G}$, and thus \n$\\bar{G}$ is Cohen-Macaulay (see also \\cite{VV})\nand $2=\\br(I)=a(\\bar{G})+\\dim A=a(\\bar{G})+2$. \nThus $a(\\bar{G})=0$, as required. \n\n\\par \\vspace{2mm} \n\\noindent $(5) \\Longrightarrow (1):$ \nSince $\\bar{G}$ is Cohen-Macaulay, \nwe have $\\br(I)=a(\\bar{G})+\\dim A=0+2=2$. \\qed\n\\end{proof}\n\n\\par \nWe notice that if $A$ is \\textit{not} a rational singularity, then \nelliptic ideals always exist. \n\n\\begin{prop} \\label{exist} Let $(A,\\mathfrak m)$ be a two-dimensional excellent \nnormal local domain containing an algebraically closed field \n$k= A\/\\mathfrak m$ \n and let $I \\subset A$ be an $\\mathfrak m$-primary integrally closed ideal \nwhich is not a $p_g$-ideal. Then there exists a positive integer $n$ such \nthat $\\overline{I^n} $ is an elliptic ideal. \nIn particular, if $A$ is not a rational singularity, then for any \n$\\mathfrak m$-primary integrally closed ideal $I$ of $A$, \nthen $\\overline{I^{n}}$ is either a $p_g$-ideal or \nan elliptic ideal for every $n \\ge p_g(A)$. \n\\end{prop}\n\n\\begin{proof} \nLet $n$ be a positive integer such that \n$\\ell_A(A\/\\overline{I^{n}})=\\bar{P}_I(n-1)$. Since the integral closure of \n$(\\overline{I^n})^{p}$ coincides with \n$\\overline{I^{n p}}$ for $p$ large, we have \n\\[\n\\bar{e}_0(I^{n})= n^2 \\bar{e}_0(I) ; \\ \\ \n\\bar{e}_1(I^{n}) = n \\bar{e}_1(I) + {n \\choose 2} \\bar{e}_0(I); \\ \\ \n\\bar{e}_2(I^n)= \\bar{e}_2(I). \n\\]\nAfter substituting the $\\bar{e}_i(I^n)$'s with the corresponding \nexpressions in terms of the $\\bar{e}_i(I)$'s we conclude that\n\\begin{eqnarray*}\n\\bar{e}_2(I^n) - \\bar{e}_1(I^n) + \\bar{e}_0(I^n)-\\ell_A(A\/\\overline{I^n})\n&=& \\bar{e}_0(I) {n+1 \\choose 2} - \\bar{e}_1(I) n + \\bar{e}_2(I) \n-\\ell_A(A\/\\overline{I^n}) \\\\\n&=& \\bar{P}_I(n-1) -\\ell_A(A\/\\overline{I^n}) =0.\n\\end{eqnarray*}\nSince $I$ is a not $p_g$-ideal, then $\\bar{e}_2(I^n)= \\bar{e}_2(I) >0. $ \nHence, by Theorem \\ref{nr2}, then $\\overline{I^n}$ is an elliptic ideal. \n\\qed\n\\end{proof} \n\n\\par \nWe denote by $\\mathfrak{M} = \\mathfrak m + \\bar{\\mathcal R}_{+} $ the homogeneous maximal \nideal of $\\bar{\\mathcal R}$. \nAs usual we say that $\\bar{\\mathcal R}$ is $($FLC$)$ if \n$\\ell_A(H^i_{\\mathfrak{M}} (\\bar{\\mathcal R}))< \\infty $ for every $i \\le \\dim A=2$. \n\n\\begin{prop} \\label{H} \nAssume $I$ is an elliptic ideal, then \n$\\bar{\\mathcal R}$ is $($FLC$)$ but not Cohen-Macaulay with \n\\[\nH^2_{\\mathfrak M}(\\bar{\\mathcal R}) = [H^2_{\\mathfrak M}(\\bar{\\mathcal R})]_0\n\\cong [H_{\\mathfrak M}^2(\\bar{G})]_0. \n\\]\n\\end{prop}\n\n\\begin{proof} \nNote that $\\bar{\\mathcal R}_{\\mathfrak M}$ is a universally catenary domain which is a homomorphic image of a Cohen-Macaulay local ring. \nHence it is an (FLC) because $\\bar{\\mathcal R}$ satisfies Serre condition $(S_2)$. Thus $H_{\\mathfrak M}^0(\\bar{\\mathcal R})= H_{\\mathfrak M}^1(\\bar{\\mathcal R})=0$ and\n$H_{\\mathfrak M}^2(\\bar{\\mathcal R})$ has finite length. \n\\par \nPut $\\mathcal{N}=\\bar{\\mathcal R}_{+}$. \nThen we obtain two exact sequences of graded $\\bar{\\mathcal R}$-modules. \n\\[\n0 \\to \\mathcal{N} \\to \\bar{\\mathcal R} \\to {}_h A \\to 0, \\; \n\\]\n\\[\n 0 \\to \\mathcal{N}(1) \\to \\bar{\\mathcal R} \\to \\bar{G} \\to 0, \\; \n\\]\nwhere ${}_h A$ can be regarded as $\\bar{\\mathcal R}\/\\mathcal{N}$ \nwhich is concentrated in degree $0$. \nOne can easily see that $H_{\\mathfrak M}^0(\\mathcal{N})=H_{\\mathfrak M}^1(\\mathcal{N})=0$, and we get \n\\begin{equation} \\label{firstEx}\n0 \\to H_{\\mathfrak M}^2(\\mathcal{N}) \\to H_{\\mathfrak M}^2(\\bar{\\mathcal R}) \\to \n{}_h H_{\\mathfrak m}^2(A) \\to H_{\\mathfrak M}^3(\\mathcal{N}) \\to H_{\\mathfrak M}^3(\\bar{\\mathcal R}) \\to 0, \\; \n\\end{equation}\n\\begin{equation} \\label{secondEx}\n0 \\to \nH_{\\mathfrak M}^2(\\mathcal{N})(1) \\to H_{\\mathfrak M}^2(\\bar{\\mathcal R}) \\to \nH_{\\mathfrak M}^2(\\overline{G}) \\to H_{\\mathfrak M}^3(\\mathcal{N})(1) \\to H_{\\mathfrak M}^3(\\bar{\\mathcal R}) \\to 0. \\; \n\\end{equation}\nFor any integer $n \\le -1$, the first exact sequence (\\ref{firstEx}) \nyields \n\\[\n0\\to [H_{\\mathfrak M}^2(\\mathcal{N})]_n \\to [H_{\\mathfrak M}^2(\\bar{\\mathcal R})]_n \\to 0. \\; \n\\] \nAlso, the second exact sequence (\\ref{secondEx}) yields \n\\[\n0=[H_{\\mathfrak M}^1(\\bar{G})]_n \\to [H_{\\mathfrak M}^2(\\mathcal{N})]_{n+1} \\to [H_{\\mathfrak M}^2(\\bar{\\mathcal R})]_n. \\; \n\\] \nThen \n$[H_{\\mathfrak M}^2(\\bar{\\mathcal R})]_{-1} \\subset [H_{\\mathfrak M}^2(\\bar{\\mathcal R})]_{-2} \n \\subset \\cdots \\subset [H_{\\mathfrak M}^2(\\bar{\\mathcal R})]_{n}=0$ for $n \\ll 0$ and thus $[H_{\\mathfrak M}^2(\\bar{\\mathcal R})]_{n}=0$ for all $n \\le -1$. \n\\par \\vspace{2mm} \nFor any integer $n \\ge 1$, the first exact sequence (\\ref{firstEx}) \nyields \n\\[\n0 \\to [H_{\\mathfrak M}^2(\\mathcal{N})]_n \\to [H_{\\mathfrak M}^2(\\bar{\\mathcal R})]_n \\to 0. \\; \n\\] \nMoreover, as $a(\\bar{G})=0$, we have \n\\[\n[H_{\\mathfrak M}^2(\\mathcal{N})]_{n+1} \\to [H_{\\mathfrak M}^2(\\bar{\\mathcal R})]_n \\to \n[H_{\\mathfrak M}^2(\\bar{G})]_n =0 \\; \\text{(ex)}. \n\\]\nHence we get $[H_{\\mathcal{M}}^2(\\bar{\\mathcal R})]_n=0$ for all $n \\ge 1$. \n\\par \nSince $a(\\bar{\\mathcal R})=-1$, we have \n$[H_{\\mathfrak M}^3(\\mathcal{N})]_1 \\cong [H_{\\mathfrak M}^3(\\bar{\\mathcal R})]_1=0$. \nHence we get \n\\[\nH_{\\mathfrak M}^2(\\bar{\\mathcal R})=\n[H_{\\mathfrak M}^2(\\bar{\\mathcal R})]_0 \\cong [H_{\\mathfrak M}^2(\\bar{G})]_0, \n\\]\nas required. \\qed\n\\end{proof}\n\n\\begin{cor} \\label{cohomology} \nLet $(A,\\mathfrak m)$ be a two-dimensional excellent normal local domain and let \n$I \\subset A$ be an $\\mathfrak m$-primary integrally closed ideal. \n\\par \\noindent\nThen $I$ is an elliptic ideal if and only if \n$0 \\ne H^2_{\\mathfrak M}(\\bar{\\mathcal R}) = [H^2_{\\mathfrak M}(\\bar{\\mathcal R})]_0 \\hookrightarrow H_{\\mathfrak m}^2(A)$, where the last map is induced from the natural surjection \n$\\bar{\\mathcal R} \\to {}_h A = \\bar{\\mathcal R}\/\\bar{\\mathcal R}_{+}$. \n\\end{cor} \n\n\\begin{proof} Assume $I$ is an elliptic ideal, then from the proof of Proposition \\ref{H} and Theorem \\ref{nr2} \nwe conclude our assertions. \nConversely, by our assumption, we can conclude that \n$\\bar{G}(I)$ is Cohen-Macaulay with $a(\\bar{G}(I))=0$ \nby a similar argument as in the proof of Proposition \\ref{H}. \nHence $I$ is an elliptic ideal by Theorem \\ref{nr2}. \\qed\n\\end{proof} \n\n\\par \nFor a cycle $C>0$ on $X$, we denote by $\\chi(C)$ the Euler \ncharacteristic of $\\mathcal O_C$. \n\n\\begin{defn} \\label{fund}\nLet $Z_f$ denote the \\textit{fundamental cycle}, namely, \nthe non-zero minimal anti-nef cycle on $X$. \nThe ring $A$ is called \\textit{elliptic} if $\\chi(Z_f)=0$. \n\\end{defn}\n\n\\par \nThe following result follows from Theorem \\ref{nr2} and \\cite{Ok}, \nTheorem $3.3$. \n\n\\begin{cor} \\label{elliptic}\nIf $A$ is an elliptic singularity, then for every integrally closed ideal $I \\subset A$ the following facts hold:\n\\begin{enumerate}\n\\item $\\bar{G}(I)$ is Cohen-Macaulay with $a(\\bar{G}(I))\\le 0$.\n\\item $I$ is elliptic or a $p_g$-ideal.\n\\end{enumerate}\nSince there always exists an ideal $I$ with $q(I)=0$, we have \n$\\bar r(A)=2$. \n\\end{cor}\n\n\\par\nThe result above gives some evidence about a positive answer to the following question:\n \n\\begin{quest} \\label{ell} Assume $\\bar r(A) =2$, \nis it true that $A$ is an elliptic singularity?\n\\end{quest}\n\n\\par \nWe can give a positive answer to Question \\ref{ell} if \n$ \\bar{e}_2(I) \\le 1$ for all integrally closed $\\mathfrak m$-primary ideals. \n In the following result we describe the integrally closed \n $\\mathfrak m$-primary ideals satisfying this minimal condition. \n\n\\begin{thm}\\label{e_2=1}\nLet $(A,\\mathfrak m)$ be a two-dimensional excellent normal local domain \nover an algebraically closed field. \nLet $I \\subset A$ be an $\\mathfrak m$-primary integrally closed ideal, \nand let $Q$ be a minimal reduction of $I$. \nPut $\\bar{G}=\\bar{G}(I)$ and $\\bar{\\mathcal R}=\\bar{\\mathcal R}(I)$. \nThen the following conditions are equivalent$:$\n\\begin{enumerate}\n \\item $\\bar{r}(I)=2$ and $\\ell_A(\\overline{I^2}\/Q I) =1$. \n \\item $q(I)=q(\\infty I)=p_g(A)-1$. \n \\item $\\bar{e}_2(I) =1$.\n \\item $\\bar{e}_1(I) = e_0(I) -\\ell_A( A\/I) + 1$ and $\\nr(I)=\\br(I)$. \n \\item $\\bar{G}$ is Cohen-Macaulay with $a(\\bar{G})=0$ and \n $\\ell_A([H^2_{\\mathfrak M}(\\bar{G})]_0)=1$. \n\\end{enumerate}\nWhen this is the case, $I$ is said to be a \\textbf{strongly elliptic ideal} \nand $\\bar{\\mathcal R}$ is a Buchsbaum ring with \n$\\ell_A(H^2_{\\mathfrak M}(\\bar{\\mathcal R})) = 1$. \n\\end{thm}\n\n\\begin{proof}\n$(1) \\Longrightarrow (2):$ By Theorem \\ref{nr2}, we have \n$p_g(A) > q(I)=q(\\infty I)$. \nIn particular, $q(2 I)=q(I)$. \nBy Proposition \\ref{eq:q(nI)}(1), $p_g(A)-q(I)=\\ell_A(\\overline{I^2}\/QI)=1$. \nConversely $(2) \\Longrightarrow (1) $ again by Proposition \\ref{eq:q(nI)}. \n\n\\par \\vspace{2mm} \n$(2) \\Longrightarrow (3):$ \nBy Proposition \\ref{p:normalHP}(4), we have \n\\[\n\\bar{e}_2(I)=p_g(A)-q(I)=1. \n\\]\n\n\\par \\vspace{2mm} \n$(3) \\Longrightarrow (2):$ \nSince $p_g(A)-q(\\infty I)=\\bar{e}_2(I)=1$ by assumption, we have \n$p_g(A)-1 = q(\\infty I) \\le q(I) \\le p_g(A)$. \nIf $q(I)=p_g(A)$, then $I$ is a $p_g$-ideal and thus $\\bar{e}_2(I)=0$. \nThis is a contradiction. Hence $q(\\infty I) = q(I)=p_g(A)-1$, as required. \n\n\\par \\vspace{2mm} \n$(1),(3) \\Longrightarrow (4):$ \nIt follows from Theorem \\ref{nr2} $(1)\\Longrightarrow (3)$ and the fact that $1< \\nr(I) \\le \\br(I)=2. $\n\n\\par \\vspace{2mm} \n$(4) \\Longrightarrow (1):$\nBy Proposition \\ref{eq:q(nI)}(1), we have \n\\begin{eqnarray*}\n\\ell_A(\\overline{I^2}\/QI)&=& (p_g(A)-q(I))-(q(I)-q(2I)), \\\\\n\\ell_A(\\overline{I^3}\/Q\\overline{I^2})&=& (q(I)-q(2I))-(q(2I)-q(3I)), \\\\\n\\vdots \\qquad ~ &=& ~ \\qquad \\vdots \n\\end{eqnarray*}\nBy a similar argument as in \\cite{Hun} \nand Proposition \\ref{p:normalHP}, we get\n\\begin{eqnarray*}\n\\bar{e}_2(I) &=& \\sum_{n=1}^{\\infty} n \\cdot \\ell_A(\\overline{I^{n+1}}\/Q\\overline{I^n}), \\\\[2mm] \n\\bar{e}_1(I)-\\bar{e}_0(I)+\\ell_A(A\/I)\n&=& \\sum_{n=1}^{\\infty} \\ell_A(\\overline{I^{n+1}}\/Q\\overline{I^n}). \n\\end{eqnarray*}\nThus our assumption implies $\\ell_A(\\overline{I^{n+1}}\/Q\\overline{I^n})=1$ \nfor some unique integer $n \\ge 1$. \nOn the other hand, since $\\nr(I)=\\br(I)$, we must have $n=1$. \n\n\\par \\vspace{2mm} \n$(1) \\Longrightarrow (5):$ \nSuppose (1). Then Theorem \\ref{nr2}$(1)\\Longrightarrow (5)$ \nimplies that $\\bar{G}$ is Cohen-Macaulay with $a(\\bar{G})=0$.\n\\par\nWe remark that $\\sqrt \\mathfrak M = \\sqrt {\\bar G_+} $ in $\\bar G, $ hence by \\cite[Proposition 3.1]{MSV}, we have \n$[H_{\\mathfrak M}^2(\\bar G)]_0 \\cong \\overline{I^2}\/QI \\cong A\/\\mathfrak m$\nhas length $1$. \nIn particular by Proposition \\ref{H}, $H_{\\mathfrak M}^2(\\bar{\\mathcal R})$ \nbecomes a $A\/\\mathfrak m$-vector space \nand thus $\\bar{\\mathcal R}$ is Buchsbaum. \n\n\\par \\vspace{2mm} \n$(5) \\Longrightarrow (1):$ \nBy Theorem \\ref{nr2}$(5) \\Longrightarrow (1)$, we have $\\br(I)=2$. \nAlso, $\\ell_A(\\overline{I^2}\/QI)=\\ell_A([H_{\\mathfrak M}^2(\\bar{G})]_0)=1$. \\qed\n\\end{proof}\n\n \n\\par\nIt is clear that if $I$ is a strongly elliptic ideal, then $I$ is an elliptic ideal. \nIn some cases they are equivalent. \nNotice that the converse is \\textit{not} true in general. \nFor instance, let $A=k[[x^2,y^2,z^2,xy,xz,yz]]\/(x^4+y^4+z^4)$. \nThen $A$ is a $2$-dimensional normal local domain with the maximal ideal \n$\\mathfrak m=(x^2,y^2,z^2,xy,xz,yz)$. Then $\\mathfrak m$ is a normal ideal and \n$Q=(x^2,y^2)$ is a minimal reduction of $\\mathfrak m$ with $\\mathfrak m^3=Q\\mathfrak m^2$. \nMoreover, $\\br(\\mathfrak m)=r(\\mathfrak m)=2$ and $\\ell_A(\\mathfrak m^2\/Q\\mathfrak m)=3$ imply \nthat $\\mathfrak m$ is an elliptic ideal but not a strongly elliptic ideal. \n\nNotice that (1) is equivalent to (3) follows also from \\cite{It3}.\n\n\\begin{prop} \\label{str} Let $(A,\\mathfrak m)$ be a two-dimensional Gorenstein \nexcellent normal local domain. Then $\\mathfrak m$ is an elliptic ideal if and only if \n$\\mathfrak m$ is a strongly elliptic ideal. \n\\end{prop} \n\n\\begin{proof} \nAssume $\\mathfrak m$ is an elliptic ideal and $Q$ be\nits minimal reduction. Since $\\bar{r}(\\mathfrak m)=2$, \n$\\mathfrak m \\overline{\\mathfrak m^2} \\subset Q$ and we have \n$\\overline{\\mathfrak m^2}\/Q\\mathfrak m \\cong (\\overline{\\mathfrak m^2}+Q)\/Q \n\\hookrightarrow A\/Q$, whose image is contained in $(Q:\\mathfrak m)\/Q$.\nSince the latter has length $1$, $\\ell_A (\\overline{\\mathfrak m^2}\/Q\\mathfrak m) =1$\nand $\\mathfrak m$ is strongly elliptic.\n\\end{proof} \n\n\n\\begin{exam} \\label{Brieskorn}\nLet $A=\\mathbb{C}[[x,y,z]]\/(x^a+y^b+z^c)$ be a Brieskorn hypersurface, \nwhere $2 \\le a \\le b \\le c$. \nThen: \n\\begin{enumerate}\n\\item $\\mathfrak m$ is a $p_g$-ideal if and only if $(a,b)=(2,2),(2,3)$. \n\\item $\\mathfrak m$ is an elliptic ideal (equivalently strongly elliptic) if and only if \n\\[\n(a,b)=(2,4),(2,5),(3,3),(3,4). \n\\]\n\\end{enumerate}\nIn particular, if $p \\ge 1$ and $(a,b,c)=(2,4,4p+1)$, then $p_g(A)=p$ and \n$\\mathfrak m$ is a (strongly) elliptic ideal. \nIt follows from \\cite[Theorem 3.1, Proposition 3.8]{OWY4}. \n\\end{exam} \n\n\\begin{exam} \nProposition \\ref{str} does not hold if $I \\neq \\mathfrak m$. \nLet $A$ be any two-dimensional excellent normal local domain\nwith $p_g(A) > 1$. Then there exist \nalways integrally closed ideals $I$ with $q(I)=0$. \nSince $q(I)=q(2I)=0$, $\\bar{r}(I) =2$ and $\\bar{e}_2(I) = p_g(A)$. \nThus \\ref{str} does not hold for such $I$. \n\\end{exam} \n\n\\par \nWe recall that an excellent normal local domain for which every integrally closed $\\mathfrak m$-primary ideal is a $p_g$-ideal, is a rational singularity ($p_g(A)=0$). This result suggests to study the next step. \n\n \n\\begin{defn}[\\textrm{e.g. \\cite{Y}}] An excellent normal \nlocal domain $A$ is a {\\it{strongly elliptic singularity}} if $p_g(A)=1$. \n\\end{defn} \n\n\\par\nNote that any strong elliptic singularity is an elliptic singularity. \nThe following result characterizes algebraically the strongly \n elliptic singularities. \n\n\\begin{thm} \\label{pg=1} Let $(A,\\mathfrak m)$ be a two-dimensional excellent \nnormal local domain containing an algebraically closed field $k= A\/\\mathfrak m$ \n and assume that $p_g(A)>0$. \nThe following facts are equivalent:\n\\begin{enumerate}\n\\item $A$ is a strongly elliptic singularity. \n\\item Every integrally closed ideal of $A$ is \neither a $p_g$-ideal or a strongly elliptic ideal.\n\\end{enumerate}\n\\end{thm}\n\n\\begin{proof} \nIt depends by the fact that always there exists an integrally closed ideal $I$ of $A$ such that $q(I)=0$.\nThus $p_g(A)= \\bar{e}_2(I)$. \\qed\n\\end{proof} \n\n\\par\nIf $A$ is a rational singularity, then every integrally closed \n$\\mathfrak m$-primary ideal is normal. \nThis is not true if $A$ is an elliptic singularity, \neven if we assume $A$ is a strongly elliptic singularity. \n \n\\begin{ex} \\label{exnormal} \n\\begin{enumerate}\n\\item Let $A = k[ X,Y,Z]\/ (X^3 + Y^3 + Z^3), $ then $A$ is Gorenstein, \n$p_g(A) = 1$ and the maximal ideal $\\mathfrak m$ is normal. \nIf we consider $I = (x,y, z^2), $ then $I^2$ is not normal.\n \\item Cutkosky showed that if \n$A= \\mathbb{Q}[[ X,Y,Z]]\/ (X^3 +3 Y^3 +9 Z^3)$, \n($\\mathbb{Q}$ rational numbers), \nthen for every integrally closed ideal $I \\subset A$, \n$I^2 $ is also integrally closed and hence normal. \nThis is because the elliptic curve does not have \nany $\\mathbb{Q}$-rational point.\n\\item Let $A=k[x,y,z]\/(x^2+y^4+z^4)$, $I=\\mathfrak m=(x,y,z)$, $Q=(y,z)$. \nThen $p_g(A)=1$ and $ \\overline{\\mathfrak m^n}= x(y,z)^{n-2}+\\mathfrak m^n$ \nfor every $n \\ge 2$. \n\\end{enumerate}\n\\end{ex} \n \n\n\\begin{prop} \\label{normal} \nLet $(A,\\mathfrak m)$ be a two-dimensional excellent normal local domain \ncontaining an algebraically closed field $k= A\/\\mathfrak m. $\nAssume that $I$ is a strongly elliptic ideal. \nThen the following conditions are equivalent$:$\n\\begin{enumerate}\n\\item $\\overline{I^2}=I^2$. \n\\item $\\overline{I^n}=I^n$ for some $n \\ge 2$. \n\\item $\\overline{I^n}=I^n$ for every $n \\ge 2$. \n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof} By Theorem \\ref{e_2=1} (1), we have \n$\\ell_A(\\overline{I^2}\/QI)=1$ and \n$\\overline{I^n} = Q \\overline{I^{n-1}}$ for $n\\ge 3$. \nHence if $I^2 = \\overline{I^2}$, then \n$I^n = \\overline{I^n}$ for all $n\\ge 2$. \\par\n\\par \nConversely, assume that $I^2 \\ne \\overline{I^2}$. \nSince $\\ell_A(\\overline{I^2}\/QI)=1$, we should have \n$I^2 = QI$. This implies that $G(I) := \n\\oplus_{n\\ge 0} I^n\/ I^{n+1}$ is Cohen-Macaulay with \n$a(G(I)) = -1$ ([\\cite{VV}, Proposition 2.6] and \\cite{GS}) and hence \n\\[\n\\ell_A( A\/ I^{n+1}) = \n e_0(I) \\genfrac{(}{)}{0pt}{0}{n+2}{2}\n- e_1(I)\\genfrac{(}{)}{0pt}{0}{n+1}{1} \n\\] \nwith $e_0(I) = \\bar e_0(I)$ and $e_1(I) = e_0(I) - \\ell_A(A\/I)$.\n\\par \nOn the other hand, by Theorem \\ref{nr2} and Corollary $\\ref{e_2=1}$, we have \n\\begin{eqnarray*}\n\\ell_A(A\/\\overline{I^{n+1}})&=& \n\\bar{P}_I(n)=\\overline{e}_0(I){n+2 \\choose 2} - \\overline{e}_1(I){n+1 \\choose 1} + \\overline{e}_2(I) \\\\\n&=& e_0(I){n+2 \\choose 2} - (e_0(I)-\\ell_A(A\/I)+1){n+1 \\choose 1} +1 \\\\\n&=& \\ell_A(A\/I^{n+1})-n.\n\\end{eqnarray*}\nThis implies that $I^n \\ne \\overline{I^n}$ for all $n\\ge 2$. \\qed\n\\end{proof}\n\n\\par \nWe can characterize the normal ideals in a strongly elliptic singularity. \nBefore showing the results, let us recall some definitions and basic facts on cycles and a vanishing theorem for elliptic singularities.\nIn the following, $A$ is an elliptic singularity \nand $X$ is a resolution of $\\Spec(A)$. \n\\par\nFor a cycle $C>0$ on $X$, we denote by $\\chi(C)$ \nthe Euler characteristic $\\chi(\\mathcal O_C)=h^0(\\mathcal O_C)-h^1(\\mathcal O_C)$. \nThen $p_a(C):=1-\\chi(C)$ is called the {\\em arithmetic genus} of $C$.\nBy the Riemann-Roch theorem, we have $\\chi(C)=-(K_X+C)C\/2$, where $K_X$ is the canonical divisor on $X$. \nFrom this, if $C_1, C_2>0$ are cycles, we have $\\chi(C_1+C_2)=\\chi(C_1)+\\chi(C_2)-C_1C_2$. \nFrom the exact sequence \n\\[\n0\\to \\mathcal O_{C_2}(-C_1) \\to \\mathcal O_{C_1+C_2} \\to \\mathcal O_{C_1} \\to 0\n\\]\nwe have $\\chi(\\mathcal O_{C_2}(-C_1))=-C_1C_2+\\chi(C_2)$. \n\\par\nIf $A$ is elliptic, then there exists a unique cycle $E_{min}$, \ncalled the {\\em minimally elliptic cycle}, such that $\\chi(E_{min})=0$ and\n$\\chi(C)>0$ for all cycles $00$ on $X$ and \n$C\\ge E_{min}$ if $\\chi(C)=0$.\n\\end{prop}\n\n\\par \nLet us recall that the fundamental cycle $Z_f$ can be computed \nvia a sequence of cycles:\n\\[\nC_0:=0, \\quad C_1=E_{j_1}, \\quad C_i=C_{i-1}+E_{j_i}, \\quad C_m=Z_f,\n\\]\nwhere $E_{j_1}$ is an arbitrary component of $E$ and $C_{i-1}E_{j_i}>0$ \nfor $2\\le i \\le m$.\nSuch a sequence $\\{C_i\\}$ is called a \n {\\em computation sequence} for $Z_f$. \n It is known that $h^0(\\mathcal O_{C_i})=1$ for $1 \\le i\\le m$ \n (see \\cite[p.1260]{La}).\n\n\\par\nThe following vanishing theorems are essential in our argument.\n\n\\begin{thm}[R{\\\"o}hr {\\cite[1.7]{Rh}}] \\label{t:rohr}\nLet $L$ be a divisor on $X$ such that $LC>-2\\chi(C)$ \nfor every cycle $C>0$ which occurs in a computation sequence \nfor $Z_f$.\nThen $H^1(\\mathcal O_X(L))=0$.\nIf $A$ is rational, then the converse holds, too.\n\\end{thm}\n\n\\par \nFrom Theorem \\ref{t:rohr} and Proposition \\ref{p:ellchi}, \nwe have the following.\n\n\\begin{cor}\\label{c:ellv}\nAssume that $A$ is an elliptic singularity. \nLet $L$ be a nef divisor on $X$ such that $L E_{min}>0$.\nThen $H^1(\\mathcal O_X(L))=0$.\n\\end{cor}\n\n\n\\begin{prop}\\label{p:c1}\nAssume that $A$ is an elliptic singularity \nand $D$ the minimally elliptic cycle on $X$. \nLet $F$ be a nef divisor on $X$.\nIf $FD>0$, then $H^1(\\mathcal O_X(F-D))=0$, \nand from the exact sequence \n$0 \\to \\mathcal O_X(F-D) \\to \\mathcal O_X(F) \\to \\mathcal O_D(F) \\to 0$, \nthe restriction map\n$H^0(\\mathcal O_X(F))\\to H^0(\\mathcal O_D(F))$ is surjective.\n\\end{prop}\n\n\n\\begin{proof} \nIf $F-D$ is nef, since $(F-D)D>0$, \nwe have $H^1(\\mathcal O_X(F-D))=0$ by Corollary \\ref{c:ellv}.\nAssume that $F-D$ is not nef.\nAs in \\cite[1.4]{G}, we have a sequence $\\{D_i\\}$ of cycles such that\n\\[\nD_0=D, \\ \\ D_i = D_{i-1} + E_{j_i}, \\; (F-D_{i-1}) E_{j_i}<0\\; \n(1\\le i \\le s), \\ \\ F-D_{s} \\text{ is nef.}\n\\]\nSince $F-Z_f$ is nef, $(F-D_{i-1}) E_{j_i}<0$ implies $D_{i-1}E_{j_i}>0$\n and $D\\le Z_f$, we see that $D_s\\le Z_f$ and $D_s$ occurs \n in a computation sequence for $Z_f$.\nThen the equalities \n$\\chi(D)=\\chi(D_s)=0$ and $\\chi(D_i) = \\chi(D_{i-1}) \n+\\chi( E_{j_i})-D_{i-1} E_{j_i}$ \nimply that $F E_{j_i}=0$, $D_{i-1} E_{j_i}=1$, \nand $h^j(\\mathcal O_{E_{j_i}}(F-D_{i-1}))=0$ for $j=0,1$ and $1\\le i \\le s$.\nSince \n\\[\n0\\le \\chi(D_s+D)=\\chi(D_s)+\\chi(D)-DD_s=-DD_s,\n\\]\nwe have $(F-D_s)D>0$.\nTherefore, from the exact sequence\n\\[\n0 \\to \\mathcal O_X(F-D_i) \\to \\mathcal O_X(F-D_{i-1})\n\\to \\mathcal O_{E_{j_i}}(F-D_{i-1}) \\to 0,\n\\]\nwe obtain $H^1(\\mathcal O_X(F-D))=H^1(\\mathcal O_X(F-D_s))=0$. \\qed\n\\end{proof}\n\n\\begin{thm}[{\\cite[2.7]{FT}}] \\label{t:FT}\nLet $C$ be a Cohen-Macaulay projective scheme \nof pure dimension $1$, and let $\\mathcal F$ be a rank $1$ \ntorsion-free sheaf on $C$.\nAssume that $\\deg \\mathcal F|_{W}:= \\chi(\\mathcal F|_{W})-\\chi(W) > -2\\chi(W)$ \nfor every subcurve $W\\subset C$. \nThen $H^1(\\mathcal F)=0$. \n\\end{thm}\n\n\\par\nTo show the normality of an ideal $I$, the following is essential. \n\n\\begin{prop} \\label{p:c2}\nLet $\\mathcal L_1$ and $\\mathcal L_2$ be nef invertible sheaves \non the minimally elliptic cycle $D$ such that \n$d_i:=\\deg \\mathcal L_i \\ge 3$ for $i=1,2$.\nThen the multiplication map\n\\[\n\\gamma\\: H^0(\\mathcal L_1)\\otimes H^0(\\mathcal L_2) \\to \nH^0(\\mathcal L_1\\otimes \\mathcal L_2)\n\\]\nis surjective.\n\\end{prop}\n\n\\begin{proof}\nFirst, note that $\\chi(W)>0$ for any cycle $0{\\alpha}>> \nH^0(\\mathcal O_X(-2Z)) \\\\\n@VVV @VVV \\\\\nH^0(\\mathcal O_D(-Z))\\otimes H^0(\\mathcal O_D(-Z)) @>>> H^0(\\mathcal O_D(-2Z)),\n\\end{CD}\n\\]\nwhere at least the maps other than $\\alpha$ are surjective.\nBy Proposition \\ref{p:c1} and its proof, we have \n$I_{2Z}=I^2+H^0(\\mathcal O_X(-2Z-D))$, \n$H^0(\\mathcal O_X(-2Z-D))=H^0(\\mathcal O_X(-2Z-D_s))$, \nand $-(Z+D_s)D\\ge -ZD\\ge 3$.\nWe have as above a surjective map\n\\[\nH^0(\\mathcal O_D(-Z))\\otimes H^0(\\mathcal O_D(-Z-D_s)) \n\\to H^0(\\mathcal O_D(-2Z-D_s))\n\\]\nand $H^0(\\mathcal O_X(-2Z-D_s)) \\subset \nI H^0(\\mathcal O_X(-Z-D_s))+H^0(\\mathcal O_X(-2Z-D_s-D))$.\nFrom these arguments, for $m>0$, we have \n$I_{2Z}\\subset I^2+H^0(\\mathcal O_X(-2Z-mD))$. \nWe denote by $H(m)$ the minimal anti-nef cycle on $X$ \nsuch that $H(m)\\ge 2Z+mD$. \nThen $H^0(\\mathcal O_X(-2Z-mD))=H^0(\\mathcal O_X(-H(m)))$, \nand for an arbitrary $n\\in \\mathbb Z_{+}$ \nthere exists $m(n)\\in \\mathbb Z_{+}$ such that $H(m(n))\\ge nE$. \nTherefore, $H^0(\\mathcal O_X(-2Z-mD))\\subset I^2$ \nfor sufficiently large $m$, and we obtain $I_{2Z}=I^2$.\n\\qed \n\\end{proof} \n\n\\begin{rem}\\label{r:ZD3}\nAssume that $A$ is elliptic.\nIt follows from Proposition \\ref{p:Cq} and Corollary \\ref{c:ellv} that \n$q(I)=0$ if and only if $ZD\\ne 0$.\nBy an argument similar to the proof of Theorem \\ref{I2}, \nwe can prove that if $ZD\\ne 0$, then $I=I_Z$ is normal \nif and only if $-ZD\\ge 3$.\n\\end{rem}\n\n\n\\par \nFor elliptic ideals in an elliptic singularity (not strongly elliptic), \nRemark \\ref{r:ZD3} cannot be applied \nbecause the condition $ZD \\ne 0$ does not hold in general. \nNext example shows that the condition $0 < -ZD <3$ is \nnot necessary for $I_Z$ being not normal.\n\n\\begin{ex} \\label{noti}\nSuppose that $p\\ge 1$ be an integer.\nLet $ A=k[x,y,z]\/(x^2+y^3+z^{6(p+1)})$ \nand assume that $X$ is the minimal resolution. \nThen $E$ is a chain of $p+1$ nonsingular curves \n$E_0, E_1, \\dots, E_p$, where $g(E_0)=1$, $E_0^2=-1$, $g(E_i)=0$, \n$E_i^2=-2$, $E_{i-1}E_i=1$ for $1\\le i \\le p$ and \n$E_iE_j =0$ if $|i-j| \\ge 2$. \nIt is easy to see that $A$ is elliptic and $E_0$ is \nthe minimally elliptic cycle.\nFurthermore, $\\mathfrak m$ is a $p_g$-ideal and $p_g(A)=p+1$ \nby \\cite[3.1, 3.10]{OWY4}.\nSince $A$ is not strongly elliptic, \nthere is a non-$p_g$-ideal $I_Z$ such that $-ZE_0=0$ \n(see Theorem \\ref{pg=1} and Proposition \\ref{p:Cq}).\nLet $W=\\sum_{i=0}^p(p+1-i)E_i$. \nThen $-W\\sim K_X$ and the exceptional part of \nthe divisors $\\di_X(x)$, $\\di_X(y)$, $\\di_X(z)$ \nare $3W$, $2W$, $E$, respectively.\nFor $1\\le n \\le p+1$, \nlet $D_n=\\gcd(nE, W):=\\sum_{i=0}^p\\min(n,p+1-i)E_i$. \n(Our cycle $D_n$ coincides with $C_{n-1}$ in \\cite[2.6]{Ok}.) \nThen $\\mathcal O_X(-2D_n)$ is generated (cf. \\cite[3.6 (4)]{Ok}) \nand $D_n^2=-n$.\nLet $I_n=I_{2D_n}$. Since the cohomological cycle of \n$(D_n)^{\\bot}$ is $W-D_n$, we have \n$q(I_n)=p_g(A)-n$ by Proposition \\ref{p:Cq}; \nnote that $-(W-D_n)\\sim K_X$ on a neighborhood of \n$\\supp(W-D_n)=E_0\\cup\\cdots\\cup E_{p-n}$.\nThen $I_n=(x,y,z^{2n})$. \nWe have $D_nE_0=0$ for $1\\le n\\le p$ and $D_{p+1}E_0=E_0^2=-1$.\nTherefore, it follows from Remark \\ref{r:ZD3} that \n$\\overline{I_{p+1}^2}\\ne I_{p+1}^2$, since $-2D_{p+1}E_0=2$.\nHowever, the condition $0<-ZE_0\\le 2$ is not necessary \nfor $I_Z$ being not normal.\nIn fact, we have $\\overline{I_{n}^2}\\ne I_{n}^2$ for all $1\\le n \\le p+1$ \nbecause $xz^n\\not \\in I_n^2$ and $(xz^n)^2\\in I_n^4$.\n\\end{ex}\n\n\\section{The existence of strongly elliptic ideals}\n\nMotivated by the fact that in every two-dimensional excellent \nnormal local domain which is not a rational singularity \nelliptic ideals always exist, it is natural to ask if it is also true for \nstrongly elliptic ideals. \nWe need some more preliminaries for proving that the answer is negative, in particular there are two-dimensional \nexcellent normal local domains with no integrally closed \n$\\mathfrak m$-primary ideals $I$ with $\\bar e_2(I)=1$. \nAssume $(A,\\mathfrak m)$ is a two-dimensional excellent normal local domain \nover an algebraically closed field. \n\\par \nLet $\\pi\\colon X \\to \\Spec A $ be a resolution of singularity with exceptional set $E=\\bigcup E_i$.\n\n\\begin{defn} \\label{cycle}\nLet $D\\ge 0$ be a cycle on $X$\nand let\n\\[\nh(D)=\\max \n\\left\\{h^1(\\mathcal O_{B}) \\,\\bigg| \\,B \\in \\sum \\mathbb Z E_i, \\; B\\ge 0, \\; \n\\supp (B)\\subset \\supp (D)\\right\\}\n\\]\nWe put $h^1(\\mathcal O_{B})=0$ if $B=0$.\nThere exists a unique minimal cycle $C\\ge 0$ such that \n$h^1(\\mathcal O_C)=h(D)$ (cf. \\cite[4.8]{Re}). \nWe call $C$ the {\\em cohomological cycle} of $D$.\nThe cohomological cycle of $E$ is denoted by $C_X$.\n\\par \nNote that $p_g(A)=h(E)$ and that \nif $A$ is Gorenstein and $\\pi$ is the minimal resolution, then the \ncanonical cycle $Z_{K_X}=C_X$ (\\cite[4.20]{Re}).\nClearly, the minimally elliptic cycle is the cohomological cycle of itself.\n\\end{defn}\n\\begin{rem}\n(1) If $C_1$ and $C_2$ are cohomological cycles of some cycles on $X$ such that $C_1 \\le C_2$ and $h^1(\\mathcal O_{C_1})0$ and \nlet $D\\ge 0$ be a reduced cycle on $X$.\nThen the cohomological cycle $C$ of $D$ is \nthe minimal cycle such that \n$H^0(X\\setminus D, \\mathcal O_X(K_X))=H^0(X,\\mathcal O_X(K_X+C))$.\nTherefore, \nif $g\\:X'\\to X$ is the blowing-up at a point in $\\supp C$ and \n$E'$ the exceptional set for $g$, then the cohomological cycle $C'$ \nof $g^*D$ satisfies that $g_*^{-1}C \\le C' \\le g^*C-E'$ and \n$h^1(\\mathcal O_{C'})=h^1(\\mathcal O_C)$; \nwe have $C'=g^*C-E'$ if $\\mathcal O_X(K_X+C)$ is \ngenerated at the center of the blowing-up.\n\\end{prop}\n\n\\begin{proof}\nLet $F>0$ be an arbitrary cycle with $\\supp(F)\\subset D$.\nBy the duality, we have $h^1(\\mathcal O_F)=h^0( \\mathcal{O}_F(K_X+F))$. \nFrom the exact sequence \n\\[\n0 \\to \\mathcal O_X(K_X) \\to \\mathcal O_X(K_X+F) \\to \\mathcal O_F(K_X+F) \\to 0\n\\]\nand the Grauert-Riemenschneider vanishing theorem, we have\n\\begin{equation}\\label{eq:h1F}\nh^1(\\mathcal O_F)=\\ell_A(H^0(X, \\mathcal O_X(K_X+F))\/H^0(X,\\mathcal O_X(K_X))).\n\\end{equation}\nOn the other hand, we have the inclusion\n\\[\nH^0(X, \\mathcal O_X(K_X+F)) \\subset H^0(X\\setminus D, \\mathcal O_X(K_X)), \n\\]\nwhere the equality holds if $F$ is sufficiently large; if the equality holds, \nwe obtain $h^1(\\mathcal O_F)=h(D)$, because the upper bound $\\ell_A(H^0(X\\setminus D, \\mathcal O_X(K_X))\/H^0(X,\\mathcal O_X(K_X)))$ for $h^1(\\mathcal O_F)$ depends only on $\\supp(D)$.\nClearly, the minimum of such cycles $F$ exists as the maximal poles of \nrational forms in $H^0(X\\setminus D, \\mathcal O_X(K_X))$.\nLet $D'=g^{-1}(D)$. Since $K_{X'}+g^*C-E'=g^*(K_X+C)$, we have \n\\[\nH^0(X', \\mathcal O_{X'}(K_{X'}+g^*C-E')) = H^0(X'\\setminus D', \\mathcal O_X(K_{X'})).\n\\]\nHence $C' \\le g^*C-E'$. The inequality $g_*^{-1}C \\le C'$ is clear.\nFrom \\eqref{eq:h1F}, we have $h^1(\\mathcal O_{C'})=h^1(\\mathcal O_C)$.\nIf $\\mathcal O_X(K_X+C)$ is generated at the center of the blowing-up, then $\\mathcal O_{X'}(K_{X'}+g^*C-E')$ has no fixed components, and the minimality of the cycle $g^*C-E'$ follows.\n\\end{proof}\n\n\\begin{defn}\nWe define a reduced cycle $Z^{\\bot}$ to be the sum of the components $E_i\\subset E$ such that $ZE_i=0$.\n\\end{defn}\n\n\\par \nFrom \\cite[3.4]{OWY2}, we have the following\n\n\\begin{prop}\\label{p:Cq}\nLet $I=I_Z$ be represented by a cycle $Z$ on $X$ and denote by $C$ the cohomological cycle of $Z^{\\bot}$. \nAssume $\\bar{r}(I)=2$, then $\\mathcal O_C(-Z)\\cong \\mathcal O_C$ and $h^1(\\mathcal O_C)=q(I)$.\n\\end{prop}\n\n\\par \nThe converse of the result above is described as follows.\n\n\\begin{prop}\nIf $C$ is the cohomological cycle of a cycle on $X$ \nwith $h^1(\\mathcal O_C)=q>0$, \nthen there exist a resolution $Y\\to \\Spec A$ and a cycle $Z>0$ on \n$Y$ such that $\\mathcal O_Y(-Z)$ is generated and $q(I_Z)=q(\\infty I_Z)=q$. \n\\end{prop}\n\n\\begin{proof}\nThere exists a cycle $W$ on $X$ such that $WE_i<0$ for all $E_i$ and\n $\\mathcal O_X(-W)$ is generated (cf. the proof of \\cite[4.5]{OWY1}). \nLet $h\\in I_W$ be a general element.\nFirst we show that there exist a resolution $Y\\to \\Spec A$ \nand a cohomological cycle $D$ on $Y$ with $h^1(\\mathcal{O}_D)=q$ \nsuch that if $Z_h$ is the exceptional part of $\\di_Y(h)$, \nthen $Z_h^{\\bot}=D_{red}$.\nWe shall obtain the resolution $Y$ from $X$ \nby taking blowing-ups appropriately as follows.\nLet $H\\subset X$ be an irreducible component \nof the proper transform of $\\di_{\\Spec A}(h)$ intersecting $C$ \nat a point $p$, and let $g\\:X'\\to X$ be the blowing-up at $p$.\nLet $C'$ be the cohomological cycle of $g^*C$.\nThen $h^1(\\mathcal O_{C'})=q$ by Proposition \\ref{p:CC}.\nIf the intersection number $C'(g_*^{-1}H)$ is positive, \nthen we take again the blowing-up at the intersection point. \nBy the property of the intersection number of curves \nand Proposition \\ref{p:CC}, taking blowing-ups in this manner, we \nobtain a resolution $Y\\to \\Spec A$ and a cohomological cycle $D$ \nwhich satisfy the conditions described above; \nin fact, for an exceptional prime divisor $F$ on $Y$, we have that \n$F\\le Z_h^{\\bot}$ if and only if $F$ does not intersect the proper \ntransform of $\\di_{\\Spec A}(h)$.\nThus it follows from \\cite[3.6]{Ok} (cf. \\cite[3.4]{OWY2}) that\n$\\mathcal O_Y(-nZ_h)$ is generated and $h^1(\\mathcal O_Y(-nZ_h))=q$ \nfor $n\\ge p_g(A)$. \nThen the cycle $Z:=p_g(A)Z_h$ satisfies the assertion.\n\\end{proof}\n\n\\begin{cor} \\label{existence}\nThere exists a strongly elliptic ideal in $A$ if and only if there exists a cohomological cycle $C$ of a cycle on a resolution \n$Y\\to \\Spec A$ such that $h^1(\\mathcal O_C)=p_g(A)-1$. \n\\end{cor}\n\n\\begin{ex} \\label{no}\nLet $C$ be a nonsingular curve of genus $g\\ge 2$ and $D$ an divisor \non $C$ with $\\deg D>0$. Let $A= \\bigoplus _{n\\ge 0} \nH^0( C, \\mathcal O_C( nD))$ and assume that $a(A)=0$. \nThen $p_g(A)=g$ and \n$A$ has no strongly elliptic ideals because any cycle $F$ on any \nresolution has $h^1(\\mathcal O_F)=0$ or $g$.\nMore precisely, if $Z E_0 = 0$, where $E_0\\subset E$ denotes the \ncurve of genus $g$, then $I_Z$ is a $p_g$-ideal; \notherwise, $q(\\infty I_Z)=0$.\n\\end{ex}\n\n\\par \nNext example shows that there are local normal Gorenstein domains that always have strongly elliptic ideals.\n\\begin{ex} \\label{si}\nLet $C$ be a nonsingular curve of genus $g\\ge 2$ and put\n\\[\nA=\\bigoplus_{n\\ge 0}H^0(\\mathcal O_C(nK_C)).\n\\]\nThen $A$ is a normal Gorenstein ring by \\cite{Wt}.\nSuppose that $f\\: X\\to \\Spec A$ is the minimal resolution.\nWe have\n\\[\np_g(A)=\\sum_{n\\ge 0}h^1(\\mathcal O_C(nK_C))=g+1\n\\]\nby Pinkham's formula \\cite{Pi}, $E\\cong C$, \n$\\mathcal O_E(-E)\\cong \\mathcal O_E(K_E)$, and $K_X=-2E$ \n(cf. \\cite[4.6]{OWY1}).\nLet $Y\\to X$ be the blowing-up at a point $p\\in E$ \nand let $E_1$ be the fiber of $p$ and $E_0$ \nthe proper transform of $E$.\nBy Proposition \\ref{p:CC}, we have $C_Y=2E_0+E_1$.\nIt follows from (b) of the theorem in \\cite[4.8]{Re} that \n$h^1(\\mathcal O_{E_0})\\le h^1(\\mathcal O_{nE_0})