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+{"text":"\\section{Introduction}\n\\label{section.1_intro}\n\nThe study of the physical properties and evolution of massive stars (M\\,>\\,8-9~M$_{\\odot}$) is crucial for many aspects of our understanding of the Universe. They play an important role in the chemodynamical evolution of the galaxies \\citepads{2012ceg..book.....M} and were key players in the epoch of reionization of the Universe (\\citeads{1999ApJ...527L...5B}; \\citeads{2000ApJ...540...39A}). They are the precursors of hyperenergetic supernovae, long-duration $\\gamma$-ray burst (see \\citeads{2012ARA&A..50..107L}, and references therein), and the recently detected gravitational wave events (e.g., \\citeads{2016PhRvL.116f1102A}; \\citeads{2017PhRvL.119p1101A}; \\citeads{2020arXiv200201950A}). Their high luminosities make them observable individually at large distances, and they are thus optimal tools for access to invaluable information about abundances and distances in galaxies at up to several megaparsec (e.g., \\citeads{2003ApJ...584L..73U}; \\citeads{2008A&A...485...41C}; \\citeads{2013ApJ...779L..20K}). Moreover, through their feedback into the interstellar medium in the form of ultraviolet radiation and stellar winds, massive stars critically affect the star formation process by both triggering the formation of new generations of stars and stopping mass accretion in the surrounding forming stars.\n\nMost massive stars are found within or are linked to young open clusters and the so-called OB associations (\\citeads{2003ARA&A..41...57L}; \\citeads{2010ARA&A..48..431P}). These stellar groupings are therefore perfect laboratories to study them. \n\n\\citetads{1978ApJS...38..309H} compiled a catalog of all known supergiants (Sgs) and O stars in associations and clusters of the Milky Way, including over 1000 objects of this type. Among the list of associations quoted in that paper, Per~OB1, which also includes the famous $h$~and~$\\chi$~Persei double cluster, clearly stands out as one of the richest. In particular, it is one of the few Galactic OB associations in which, given its age ($\\sim$\\,13\\,--\\,14\\,Myr, \\citeads{2002ApJ...576..880S}; \\citeads{2019ApJ...876...65L}), a massive star population covering a wide range of evolutionary stages can be found (e.g., it harbors 23 red Sgs and several dozen blue Sgs). In addition, it is also relatively close to us (d\\,$\\sim$\\,2.2\\,--\\,2.4\\,kpc, \\citeads{2018A&A...616A..10G}; \\citeads{2019MNRAS.486L..10D}) and is characterized by a moderate extinction (E(B-V)$\\sim$0.6, \\citeads{2002ApJ...576..880S}). This unique combination of characteristics makes Per~OB1 a very interesting testbed for the study of a large interrelated population of evolved massive stars from an evolutionary point of view.\n\nPer~OB1 has attracted the attention of the astrophysical community for many years and has been the subject of studies from many different fronts. We highlight the investigation of how the association could have been formed (\\citeads{2008ApJ...679.1352L}); the membership of stars to the association (\\citeads{1970ApJ...160.1149H}; \\citeads{1978ApJS...38..309H}; \\citeads{1992A&AS...94..211G}; \\citeads{2008ApJ...679.1352L}; \\citeads{2009MNRAS.400..518M}) and, in particular, to $h$~and~$\\chi$~Persei (\\citeads{2002PASP..114..233U}; \\citeads{2010ApJS..186..191C}); the characterization of the kinematics of the region (\\citeads{2017MNRAS.472.3887M}; \\citeads{2019A&A...624A..34Z}); the identification of blue Sg binaries (\\citeads{1973ApJ...184..167A}); or the spectroscopic characterization of different samples of blue stars in the region (including the determination of rotational velocities, stellar parameters and surface abundances \\citeads{1968ApJ...154..933S};  \\citeads{1988A&A...195..208L}; \\citeads{1995A&A...298..489K}; \\citeads{1996A&A...310..564K}; \\citeads{2005AJ....129..809S}; \\citeads{2019ApJ...876...65L}), also reaching the red Sg domain (\\citeads{2014ApJ...788...58G}).\n\nDespite all the information compiled about the Per~OB1 association, and particularly, $h$~and~$\\chi$~Persei, we still lack a complete homogeneous empirical characterization (that also takes environmental and kinematical information into account) of the physical and evolutionary properties of its massive star population. This is the main objective of this series of papers, which is based on a set of high-quality observations including high-resolution, multi-epoch spectroscopy (mostly gathered in the framework of the IACOB project, see \\citeads{2015hsa8.conf..576S} and references therein), and astrometric information delivered by the {\\em Gaia} mission (\\citeads{2018A&A...616A...1G}; \\citeads{2018A&A...616A...2L}). The compiled empirical information resulting from the analysis of this observational dataset will allow us to proceed in our understanding of massive star evolution, and also investigate some long-standing and new open questions in this important field of stellar astrophysics. These questions include the evolutionary status of the blue supergiants, or the effect that binarity and rotation have on the evolution of massive stars.\n\nIn this first paper, we carry out a membership analysis of a sample of 88 blue and red Sgs located within 4.5\\,deg from the center of the Per~OB1 association, and we also investigate some of its kinematical properties. In Sect.~\\ref{section.2_samobs} we present the sample of stars and the main characteristics of the compiled observations. \nIn Sect.~\\ref{section.3_rv} we describe the strategy we followed to derive reliable radial velocities (RVs). Sect.~\\ref{section.4_results} presents the results extracted from the analysis of the observations, mainly referring to parallaxes, proper motions, RV measurements, and the identification of spectroscopic signatures of binarity and other types of spectroscopic variability phenomena.\nIn Sect.~\\ref{section.5_discus} we use all these results to establish and apply our membership criteria to all stars in the sample, and we also identify outliers for each of the considered quantities, in particular, binary and runaway stars. We also analyze some global features of Per~OB1, and discuss some individual cases of interest. The main conclusions of this work and some future prospects are provided in Sect.~\\ref{section.6_summary}.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=.47\\textwidth]{figures\/Fig1.png}\n\\caption{Sky map with the complete sample of stars. Purple, blue, cyan, golden, and red symbols represent the O-, B-, A-, F-, and K- and M-type stars, respectively. This color code is the same in all the plots unless otherwise specified. The central green cross denotes the center of the Per~OB1 association taken from \\citetads{2017MNRAS.472.3887M}. The large green circle indicates a 4.5-degree circle around the center. The small green circles show the positions of $h$~and~$\\chi$~Persei. The background image, used for reference, was taken from DSS-red.}\n\\label{figure.fig1}\n\\end{figure}\n\n\n\n\n\\section{Sample definition and observations}\n\\label{section.2_samobs}\n\nIn this section, we describe the process we have followed to build the sample under study, and to compile the associated observations. The latter mainly refers to high-quality spectroscopy obtained with the FIbre-fed Echelle Spectrograph (FIES), \\citepads{2014AN....335...41T} and the High Efficiency and Resolution Mercator Echelle Spectrograph (HERMES) \\citepads{2011A&A...526A..69R} high-resolution\nspectrographs attached to the 2.56~m Nordic Optical Telescope (NOT) and the 1.2~m Mercator telescope, respectively, and astrometric and photometric data delivered by the {\\em Gaia} mission in the second data release \\citepads[DR2, ][]{2018A&A...616A...1G, 2018A&A...616A...2L, 2018A&A...616A...4E}.\n\n\n\\subsection{Sample definition}\n\\label{subsection.21_sample}\nThe final sample of targets considered for this work comprises 88 blue and red massive stars located within 4.5\\,deg from the center of the Per~OB1 association (as defined in \\citeads{2017MNRAS.472.3887M}). \nTo restrict the sample to the most massive stars, the luminosity classes (LCs) were limited to bright giants (Gs) and Sgs (LC II and I, respectively) in the case of the O- and B-type stars, and to Sgs when we refer to A- and later-type stars. In addition, the sample includes a few O and early-B Gs for which we already had available observations in the IACOB spectroscopic database (see Sect.~\\ref{subsection.22_specobs}).\n\nTable~\\ref{table.A1} summarizes the list of targets, separated and ordered by spectral type (SpT). We note that the quoted spectral classifications were carefully revised using the spectra with the best signal-to-noise ratio (S\/N) of our own spectroscopic observations (see Sect.~\\ref{subsection.22_specobs}) following the criteria explained in \\cite{Negueruela2020}, in prep.) and \\citeads{2018A&A...618A.137D}, for the case of the blue and red Sg samples, respectively.\nIn addition, Fig.~\\ref{figure.fig1} shows their location on the sky. We also indicate as a large green circle the search area of 4.5\\,deg around the center of Per~OB1, marked as a green cross. Most stars, including those from the $h$~and~$\\chi$~Persei double cluster (indicated as two small green circles), are concentrated along the diagonal of the image. In addition, our sample includes four stars lying within one degree from the center of IC\\,1805 (the Heart nebula, located in the top left corner of the figure). The top panel in Fig.~\\ref{figure.fig2} depicts the histogram of SpT of the sample, which shows that the majority of stars are B~Sgs.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=.47\\textwidth]{figures\/Fig2a.png}\n\\includegraphics[width=.47\\textwidth]{figures\/Fig2b.png}\n\\caption{Histograms by SpT separated with colors (top) and number of spectra separated by SpT and stacked (bottom).} \n\\label{figure.fig2}\n\\end{figure}\n\nTo assemble this sample of stars, we considered several bibliographic sources, including the works by \\citetads{1978ApJS...38..309H}, \\citetads{1992A&AS...94..211G}, \\citetads{2010ApJS..186..191C}, and \\citetads{2014ApJ...788...58G}. In a first step, we used the Topcat\\footnote{\\href{Topcat}{http:\/\/www.star.bris.ac.uk\/~mbt\/topcat\/}} Virtual Observatory tool to cross-match all the stars that are quoted in these four papers and fulfilled the criteria indicated above and the list of targets with spectra available in the IACOB spectroscopic database. In a second step, we tried to obtain new spectra of as many of the missing stars as possible using the NOT or Mercator telescopes (see Sect.~\\ref{subsection.22_specobs}).\n\nFrom the original lists of luminous stars in Galactic OB associations quoted in \\citetads{1978ApJS...38..309H} and \\citetads{1992A&AS...94..211G}, we found 207 targets that are located within 4.5\\,deg of the center of Per~OB1. Only 109 of these fulfill our luminosity class criteria; the rest are either dwarfs, (sub)giants, or do not have a defined luminosity class. Our sample includes 82 of these, but we miss spectra for another 12 (7 B and 5 M Sgs).\n\nWe also used the list of targets quoted in the extensive study of the stellar population of $h$~and~$\\chi$~Persei by \\citetads{2010ApJS..186..191C} to find suitable candidates. From the complete list of several ten thousand stars, only 23 were found to have luminosity classes I or II. We currently have spectra for 17 of them. Of the remaining 6 (all of them B~Sgs), one was identified previously when we cross-matched our observations with the list of targets in \\citetads{1978ApJS...38..309H} and \\citetads{1992A&AS...94..211G}. This means that we lack spectra for another 5 blue Sgs at the time of writing.\n\nLast, our sample includes all the red Sgs of those listed in \\citetads{2014ApJ...788...58G}. In summary, the sample of stars we discuss here comprises all the blue and red Sgs (except for 12 B and five M Sgs, listed at the end of Table~\\ref{table.A1} for future reference) that are quoted in the abovementioned papers and are located within 4.5\\,deg around the center of Per~OB1. Further notes on the actual completeness of our sample can be found in Sect.~\\ref{subsection.43_compl}.\n\n\n\\subsection{Spectroscopic observations}\n\\label{subsection.22_specobs}\n\nThe spectroscopic observations of the stars in the sample come from different observing runs performed between November 2010 and December 2019 using either the FIES (NOT) or the HERMES (Mercator) instruments.\n\nThe first observations, comprising an initial sample of B, A, and M Sgs in Per~OB1 selected from \\citetads{1978ApJS...38..309H}, were obtained in 2010 during an observing run of three nights with Mercator (PI. M.A. Urbaneja). The O stars in the sample were targeted by the IACOB project (P.I. S. Sim\\'on-D\\'iaz) as part of a more general objective of observing all O stars in the Northern Hemisphere up to $V_{mag}$ = 9. These observations, obtained with both HERMES and FIES, include a minimum of three epochs per target (see more details in \\citeads{2018A&A...613A..65H}; \\citeads{Holgado2019}; \\citeads{2020arXiv200505446H}). We also benefit from the multi-epoch observations available for a subsample of O and B~Sgs as gathered by the IACOB project as part of a subproject aimed at investigating line-profile variability phenomena in the OBA Sg domain and its relation with pulsational-type phenomena (see, e.g., \\citeads{2010ApJ...720L.174S}; \\citeads{2017A&A...597A..22S}; \\citeads{2018A&A...612A..40S}; \\citeads{2017A&A...602A..32A}; \\citeads{2018MNRAS.476.1234A}). The time span of these observations covers several years. We also count on multi-epoch observations of red Sgs obtained during several of our observing runs with HERMES. Last, all these observations have more recently been complemented by FIES spectroscopy obtained as part of the time granted to A. de Burgos in 2018 by the Spanish time-allocation committee, and through internal service observations performed in 2019 and 2020 by A. de Burgos. In addition, we were able to obtain a new epoch for a large fraction of stars in the sample during an observing run with Mercator in December 2019. \n\nFIES is a cross-dispersed high-resolution \\'echelle spectrograph mounted at the 2.56~m Nordic Optical Telescope (NOT), located at the Observatorio del Roque de los Muchachos on La Palma, Canary Islands, Spain. The observations made with FIES were taken with different fibers\/resolutions from R\\,$\\sim$\\,25000 to R\\,$\\sim$\\,67000, and with a wavelength coverage of 370-830\\,nm. \n\nHERMES is a fibre-fed prism cross-dispersed \\'echelle spectrograph mounted at the 1.2~m Mercator Telescope, also located at the Observatorio del Roque de los Muchachos. It provides a spectral resolution of R\\,$\\sim$\\,85000 and wavelength coverage of 377-900 nm, similar to FIES. \n\nThe FIES and HERMES spectrographs provide good mechanical and thermal stability that allows for a good precision in RV measurements. For FIES, the RV y accuracy\\footnote{http:\/\/www.not.iac.es\/instruments\/fies\/fies-commD.html} for the high-resolution fiber has been proved to be 5-10\\,m\/s, regardless of the atmospheric conditions. For the medium-resolution fiber under poor conditions, the precision reaches 150\\,m\/s. In the case of HERMES, the precision obtained for the low- and high-resolution fibers is 2.5 and 2\\,m\/s, respectively (\\citeads{2011A&A...526A..69R}). In both cases this precision is well above the precision required for this work, as we expect variations of several \\kms\\ for the blue Gs\/Sgs, and a few \\kms\\ for the red Sgs. \n\nAll the spectra were reduced using the FIESTool (\\citeads{2017ascl.soft08009S}) and HermesDRS\\footnote{http:\/\/www.mercator.iac.es\/instruments\/hermes\/drs\/} dedicated pipelines. Both pipelines provide merged wavelength-calibrated spectra. In addition, we used our own programs, implemented in IDL, to normalize the spectra and compute the heliocentric velocity to be applied to each spectrum before the associated RV was measured (see Sect.~\\ref{section.3_rv}).\n\nAs indicated above, we have multi-epoch spectroscopy for a large fraction of the stars in our sample. The bottom panel in Fig.~\\ref{figure.fig2} summarizes this characteristic of our observations, showing the histogram of the collected number of spectra per star. In addition, Table~\\ref{table.A2} quotes all those stars for which we have five or more spectra. This table includes the time span covered by the spectra, together with the total number of spectra for each of these stars, separated by SpT. It is important to remark that the cadence of the spectra taken for each star is very inhomogeneous, as they were gathered during different observing runs, as described at the beginning of this section. \n\n\n\\subsection{Photometric and astrometric data}\n\\label{subsection.23_gaiaobs}\n\nFor all the stars in the sample, Table~\\ref{table.A1} quotes the {\\em Gaia} $G_{mag}$ and $BP_{mag}$\\,-\\,$RP_{mag}$, parallaxes ($\\varpi$) and proper motions ($\\mu_{\\alpha}$,$\\mu_{\\delta}$), as well as associated errors, retrieved from {\\em Gaia} DR2. Sources in the {\\em Gaia} catalog were identified using Topcat, defining a radius threshold of 2~arcsec. \n\nWe adopted a parallax zero-point offset of $-0.03$\\,mas (see \\citeads{2018A&A...616A...2L}), which is already applied to all values quoted in Table~\\ref{table.A1} and used to generate the various figures in the paper. We note, however, that some other authors push this value up to $-0.08$\\,mas (see \\citeads{2018ApJ...862...61S}; \\citeads{2019MNRAS.486L..10D}). \n\nThe {\\em Gaia} DR2 renormalized unit weight error (RUWE) is also included in the last column of Table~\\ref{table.A1}. The value of this quantity is used to estimate the goodness of the {\\em Gaia} astrometric solution for each individual target. Following recommendations by the {\\em Gaia} team for the known issues\\footnote{https:\/\/www.cosmos.esa.int\/web\/gaia\/dr2-known-issues}, we decided to adopt a RUWE\\,=\\,1.4 to distinguish between good and bad solutions.\n\nSeven stars (or 8$\\%$ of the sample) have an associated RUWE higher than this value. Their parallaxes and proper motions are indicated in parentheses in Table~\\ref{table.A1}. Hereafter, we call them stars with \"unreliable astrometry\" or \"unreliable astrometric solution\". For all the stars with a RUWE $<$ 1.4, the top panel of Fig.~\\ref{figure.fig3} shows the $G_{mag}$ against the {\\em Gaia} error in parallax, and the bottom panel shows the {\\em Gaia} error in total proper motion against the {\\em Gaia} error in parallax. \n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=.47\\textwidth]{figures\/Fig3.png}\n\\caption{(Top) $G_{mag}$ against the {\\em Gaia} error in parallax. (Bottom) {\\em Gaia} error in total proper motion against the error in parallax. Both panels include all stars in our sample except for the seven targets with {\\em Gaia} RUWE > 1.4 (see Sect.~\\ref{subsection.23_gaiaobs}).}\n\\label{figure.fig3}\n\\end{figure}\n\nThe $G_{mag}$ of the stars in our working sample ranges between 5.1 and 9.7\\,mag. It has been shown that bright sources ($G_{mag}$ < 6) also result in unreliable astrometric solutions because of uncalibrated CCD saturation (\\citeads{2018A&A...616A...2L}). In the sample, four stars have magnitudes lower than 6, and they are discussed in detail in Sect.~\\ref{subsection.51_memb}. In order to verify the {\\em Gaia} DR2 parallaxes and proper motions for the brightest stars in the sample, we also retrieved the values provided in the {Hipparcos} (\\citeads{2007A&A...474..653V}), and TGAS (\\citeads{2015A&A...574A.115M}) catalogs; however, the results were not better. \n\nThe {\\em Gaia} errors in parallax range between 0.032, and 0.121\\,mas, while the errors in total proper motion range between 0.046, and 0.308\\,mas\/yr. Six stars have uncertainties in parallax $\\sim$0.08\\,mas or larger. The same have uncertainties in total proper motion larger than 0.18\\,mas\/yr. They are all red Sgs except for HD\\,14489 (the A~Sg in the upper right corner). The explanation for their large errors lies in the combined effect of large size and variability for the red Sgs, and the high brightness for HD\\,14489 ($G_{mag}$ = 5.1). In both cases, the {\\em Gaia} astrometric solution is affected (see Sect.~\\ref{subsection.51_memb}). \nOf the red Sgs, HD\\,14528 (in the upper right corner) has the largest errors and also a relatively high RUWE value (1.25), followed by HD\\,14489, which in comparison has a RUWE = 0.81. In particular, for HD\\,14528, we adopt the results from \\citetads{2010ApJ...721..267A} from this point on, who used the very long baseline interferometry (VLBI) technique to derive the astrometric parameters.\n\nFor the stars with RUWE $>$ 1.4, errors in parallax range between 0.086, and 0.384\\,mas with a mean of 0.170\\,mas, and errors in total proper motion range between 0.171, and 0.501\\,mas\/yr, with a mean of 0.347\\,mas\/yr. We note that as expected, all these stars have larger errors than those associated with the main concentration of stars in the bottom panel of Fig.~\\ref{figure.fig3}. \n\n\n\n\\section{Radial velocity measurements}\n\\label{section.3_rv}\n\nWe first generated various suitable lists of spectral lines, optimized for the different SpT, using information available in the Atomic Line List interface\\footnote{https:\/\/www.pa.uky.edu\/~peter\/newpage\/} \\citep{2018Galax...6...63V}, and the SpectroWeb\\footnote{\\href{SpectroWeb}{http:\/\/spectra.freeshell.org\/whyspectroweb.html}} database. Each line list comprises a few to several dozen strong (log(gf) > --0.5), unblended lines covering the full 390\\,--\\,650~nm spectral window (or 510\\,--\\,870~nm in the case of the red Sgs). \n\nFor early and mid O-type stars, a few lines of N~{\\sc iii-v} and O~{\\sc iii} were used. In addition, we also included some He~{\\sc i} lines to compensate for the lower number of available metal lines. For the late O-type stars, we added some lines of Si~{\\sc iv} and O~{\\sc ii}. The situation improves for the B and A Sgs, were a much larger sample of lines is available, including lines from Si~{\\sc ii-iv}, N~{\\sc ii-iii}, O~{\\sc ii-iii}, S~{\\sc ii-iii}, C~{\\sc ii}, Mg~{\\sc ii}, and Fe~{\\sc ii}. Last, in the case of red Sgs, we mostly used lines from Mg~{\\sc i}, Ti~{\\sc i}, Fe~{\\sc i}, Ca~{\\sc i}, Cr~{\\sc i}, Ni~{\\sc i}, and V~{\\sc i}.\n\nWe then used our own tool (developed in Python\\,3.6) to perform a RV analysis. For each star, the corresponding list of lines was selected based on its SpT. For each line, an iterative normalization of the surrounding local continuum was made. Then, each line was fit to either a Gaussian or a Gaussian plus a rotational profile, depending on the first estimate of the full width at half maximum (FWHM) of the line. The measured central wavelength was then used to calculate the RV of each individual line in the initial line list (see above). From all the identified lines we removed those with equivalent widths lower than 25\\,{m\\AA} directly before we carried out an iterative sigma clipping (using a threshold of 2\\,$\\sigma$) to remove potential poorly fit lines or incorrect identifications. The RVs of the surviving lines were then averaged, and we calculated the standard deviation of the final RV. This process was repeated for each spectrum and for each star in the sample. \n\nThe measurements of the individual RVs, together with the number of lines used for each spectrum, are listed in Table~\\ref{table.rvstable}. For O-type stars, the average number of lines is 12, the final average number of lines after sigma clipping is 6, and the typical uncertainties associated with the dispersion of RV measurements obtained after sigma clipping is $\\sim$3.9\\,\\kms. For the B-type stars, these values are 37 and 22 lines and $\\sim$0.9\\,\\kms , respectively. For A\/F-type stars, they are 42 and 32 lines, and $\\sim$0.26\\,\\kms. Finally, for the K\/M-type stars, they are 31 and 24 lines, and $\\sim$0.17\\,\\kms. This error is larger for the O-type stars for two main reasons: the first is that fewer lines are available, and the second reason is related to the broadening of the diagnostic lines, which is much larger for the O-type stars than in the cooler B, A, and red Sgs.\n\nThe RV results for the spectra with the best S\/N are shown in the last column of Table~\\ref{table.A1}. For each star, we also searched for double-lined spectroscopic binaries (SB2) by looking at different key diagnostic lines (e.g., \\ioni{He}{i}~$\\lambda$5875, \\ioni{Si}{iii}~$\\lambda$4552, \\ioni{O}{iii}~$\\lambda$5592, \\ioni{C}{ii}~$\\lambda$4267, \\ioni{and Mg}{ii}~$\\lambda$4481). \n\nWe were able to measure individual RVs for the two components in three of the five SB2. We used the spectrum of maximum separation between them. Their values are listed in Table~\\ref{table.A1}.\n\nFor each star with four or more spectra, an average RV was calculated as the mean of the RVs obtained for each individual spectrum. In addition to the associated standard deviation, the peak-to-peak amplitude of variability in RV (RV$_{\\rm PP}$) was calculated as the difference between the highest and lowest individual RVs, and its error was calculated as the square root of the sum of the squares of the their individual uncertainties. The results for the stars for which multi-epoch spectroscopy is available are listed in Table~\\ref{table.A2}. \n\nLast, we also visually inspected the line-profile variability in each star with available multi-epoch spectroscopy. By doing this we were able to identify those cases in which any detected variability is more likely due to stellar oscillations than to (single-line) spectroscopic binarity (see Sect.~\\ref{subsubsection.442_multi}) \n\n\n\n\n\\section{Results}\n\\label{section.4_results}\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=.72\\textwidth]{figures\/Fig4.png}\n\\caption{{\\em Top pannel}: Sky map of all the stars in the sample overplotted over a DSS-red image of the region. Dashed yellow line indicates the galactic plane, and the central green cross marks the center of Per~OB1 (as defined in \\citetads{2017MNRAS.472.3887M}). Green circles indicated the location of the $h$~and~$\\chi$ Persei double cluster. Colored vectors indicate the individual proper motion of each star. {\\em Middle and bottom panels}: Parallax and RV of the spectrum with the highest S\/N, respectively, for each star in the sample against their position in right ascension. Open circles and square symbols indicate stars that deviate more than 2$\\sigma$ from the mean of the distribution of parallaxes and RVs, respectively (see Sects.~\\ref{subsection.41_plx&pm} and \\ref{subsubsection.441_bestsnr}). Stars with bad astrometry (see Sect.~\\ref{subsection.23_gaiaobs}) are indicated with a plus, and no proper motion vectors are overplotted.}\n\\label{figure.fig4}\n\\end{figure*}\n\nFig.~\\ref{figure.fig4} summarizes all the compiled information on astrometry and RVs (except for the information we extracted from the multi-epoch spectroscopy, which is presented in Fig.~\\ref{figure.fig9}). The top panel of the figure shows the position of the stars in the sky, and the corresponding proper motions are indicated with arrows. For reference, we also indicate the location of the $h$~and~$\\chi$~Persei double cluster (green circles at the center of the image) and the Galactic plane (dashed yellow line).\n\nThis image is complemented with another two panels, in which the distribution of parallaxes and RVs (as derived from the best S\/N spectrum of each star) is plotted against the right ascension (middle and bottom panels, respectively). These two panels allow us to better identify the location in the sky of the outliers of both distributions, and to easily connect the information of the three investigated quantities.\n\nFrom a first visual inspection of this summary figure, it becomes clear that generally speaking, the stars in our sample (including those located in the $h$~and~$\\chi$~double cluster) belong to a connected population in terms of proper motions, parallaxes, and RVs. In addition, there is a non-negligible number of outliers that we discuss in detail in the next sections. They are potential nonmembers of the Per~OB1 association, and\/or runaway stars and binary systems.\n\n\n\\subsection{Parallaxes and proper motions}\n\\label{subsection.41_plx&pm}\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=.49\\textwidth]{figures\/Fig5.png}\n\\caption{Total proper motions against parallax for the sample of stars except for those labeled \"unreliable astrometry\" (see Sect.~\\ref{subsection.23_gaiaobs}). The 2$\\sigma$ boundaries of the distribution are shown as a rectangle. Empty colored circles show outliers of the distribution of any of the two quantities, and the associated uncertainties are overplotted. The mean and standard deviation obtained from the stars within the 2$\\sigma$ box are shown in the top right corner.}\n\\label{figure.fig5}\n\\end{figure}\n\nFigure~\\ref{figure.fig5} shows again the results for proper motions and parallaxes ($\\varpi$) from a different perspective. The central panel of the figure depicts the combined distribution of these two quantities, this time using the modulus of the proper motion ($\\mu$), defined as the square root of the sum of the squares of the proper motion in right ascension and declination. Stars labeled \"unreliable astrometric solution\" (see Sect.~\\ref{subsection.23_gaiaobs}) are excluded from this figure. \n\nMost of the stars are grouped together around $\\varpi$\\,$\\approx$\\,0.4~mas and $\\mu$\\,$\\approx$\\,1.2~mas\\,yr$^{-1}$. This is also shown in the left and bottom panels of Fig.~\\ref{figure.fig5}, where histograms of both parallax and total proper motion are shown. \n\nAn iterative 2$\\sigma$ clipping of these distributions results in $\\varpi$ = 0.40 $\\pm$ 0.07~mas, and $\\mu$ = 1.22 $\\pm$ 0.26~mas\\,yr$^{-1}$, and the identification of a total of 18 outliers (i.e., deviating more than 2$\\sigma$ from the mean of the distribution). The 2$\\sigma$ boundaries of the distribution (0.265 < $\\varpi$ < 0.540~mas, and 0.706 < $\\mu$ < 1.740~mas\\,yr$^{-1}$, respectively) and the outliers are highlighted in Fig.~\\ref{figure.fig5}. The latter are also indicated in the second and third columns of Table~\\ref{table.A4} and are discussed in Sect.~\\ref{subsection.51_memb}.\n\nThese results assume that no different local substructures exist in the region, especially in terms of parallax. To investigate this statement further, we show again in Fig.~\\ref{figure.fig6} an image of the region with the proper motions overplotted, but this time using the mean proper motion obtained by considering the 16 stars located within 15\\,arcmin from the center of $h$~and~$\\chi$~Persei, respectively, and having reliable astrometry (see the black arrow in the bottom right corner of the figure, corresponding to $\\mu_{\\alpha}\\cos{\\delta}$ = $-0.47$ and $\\mu_{\\delta}$ = $-0.99$\\,mas\\,yr$^{-1}$). \n\nThis figure is complemented with the information provided in Table~\\ref{table.radiplxpm}, where we summarize the resulting means and standard deviations of parallaxes and proper motions when the sample is divided into circular regions around the center of $h$~and~$\\chi$~Persei. The first region only includes the double cluster. The other regions extend outward by one degree each, starting at a distance of 30 arcmin from the center of the double cluster.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=.49\\textwidth]{figures\/Fig6.png}\n\\caption{Same as the top panel of Fig.~\\ref{figure.fig4}, but this time, the individual proper motion of each star is referred to the mean proper motion of 16 stars in $h$~and~$\\chi$~Persei with good astrometric solution (black arrow in the bottom right corner of the top panel).}\n\\label{figure.fig6}\n\\end{figure}\n\n\n\\begin{table}\n  \\centering\n  \\caption[]{Mean and standard deviation of parallaxes and proper motions for different groups of stars located at increased distance from the center of $h$~and~$\\chi$~Persei. Proper motions are referred to the mean of the proper motions of stars within 15 arcmin of each of the clusters and with good astrometric solution.} \n    \\label{table.radiplxpm}\n      \\begin{tabular}{cccc}\n\\hline\n\\hline\n\\noalign{\\smallskip}\nRadius [deg]  & N$_{\\rm stars}$  & $\\varpi$ [mas] &$\\mu$ [mas\/yr] \\\\  %\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n$h$~and~$\\chi$ Persei & 16 & 0.43 $\\pm$ 0.06 & 0.31 $\\pm$ 0.13 \\\\\n0.5 < R < 1.5  &19  &0.39 $\\pm$ 0.07  & 0.43 $\\pm$ 0.22 \\\\ %\n1.5 < R < 2.5  & 9  &0.34 $\\pm$ 0.04  & 0.54 $\\pm$ 0.33 \\\\ %\n2.5 < R < 3.5  & 8  &0.40 $\\pm$ 0.07  & 0.60 $\\pm$ 0.23 \\\\ %\n3.5 < R < 4.5  &14  &0.40 $\\pm$ 0.07  & 0.74 $\\pm$ 0.29 \\\\\n\\hline\n      \\end{tabular}\n\\end{table}\n\nBased on the results presented in this section, we conclude the following: There is some empirical evidence of the existence local substructures in the spacial distribution of proper motions (see further discussion in Sect.~\\ref{subsubsection.521_pm}). These subgroups of stars have a compatible distribution of parallaxes and proper motions. As a result, this justifies the decision to use the whole sample of stars to obtain the mean values and standard deviations of these two quantities to characterize this population of stars, as well as to identify potential outliers in parallax (i.e., nonmembers) and proper motion (i.e., runaway stars).\n\n\n\\subsection{Comparison with previous works}\n\\label{subsection.compa}\n\n\n\\subsubsection{Distance}\n\\label{subsection.compa_plx}\n\nWe have obtained an average value for the parallax of $\\varpi$ = 0.398 $\\pm$ 0.066\\,mas (adopting a zero offset of $-0.03$~mas). This value represents the mean of all stars in the sample with good astrometric solution that are not outliers in parallax and total proper motion.\n\nBased on the corrected computed distances to these stars from \\citetads{2018AJ....156...58B}, we obtain an average of $d$ = 2566 $\\pm$ 432\\,pc. This is compatible with the distance obtained using the inverse of our derived parallax: $d$ = 2510 $\\pm$ 415\\,pc. When we assume this distance, the projected distance extends up to $\\sim$180\\,pc for the furthest stars in the association. In particular for the stars in the double cluster used in Table~\\ref{table.radiplxpm}, we obtain a distance of $d$ = 2340 $\\pm$ 328\\,pc using the inverse of the derived parallax. \n\nThis result agrees well with previous estimates for $h$~and~$\\chi$~Persei using different approaches. In addition, it also indicates that the parallax zero-point offset correction proposed by \\citetads{2018A&A...616A...2L} is adequate.\n\nTo give some examples, \\citetads{2002PASP..114..233U} obtained an average double cluster distance of $d$ = 2014 $\\pm$ 46\\,pc using the ZAMS fitting approach. \\citetads{2010ApJS..186..191C} used main-sequence stars with a very large sample and obtained a distance to each cluster of $d_{h}$ = 2290$^{+87}_{-82}$\\,pc and $d_{\\chi}$ = 2344$^{+88}_{-85}$\\,pc. The previously mentioned work by \\citetads{2010ApJ...721..267A} estimated a distance to HD\\,14528 of $d_{h}$ = 2420$^{+110}_{-90}$\\,pc using high-precision interferometric observations. More recently, \\citetads{2018A&A...616A..10G} published mean parallaxes for a broad selection of open clusters using {\\em Gaia} DR2 including $h$~and~$\\chi$~Persei. By applying a --0.03\\,mas zero-point offset, they obtained $d_{h}$ = 2239\\,pc and $d_{\\chi}$ = 2357$^{+88}_{-85}$\\,pc. Finally, \\citetads{2019MNRAS.486L..10D} estimated the distance to $h$ Persei in $d_{h}$ = 2250$^{+160}_{-140}$\\,pc, adopting an offset of $-0.05$\\,mas for the {\\em Gaia} parallaxes. \n\nThe aim of this work is not to provide a better estimate, but to ensure that the stars selected here based on their parallax belong to the association. Only a few works provide distances to the Per~OB1 association. For instance, \\citetads{2019ApJ...882..180S} used the photometric distance and {\\em Gaia} parallaxes for a selection of O-type stars to derive a distance to the association of $d$ = 2.99\\,$\\pm$\\,0.14\\,kpc and $d$ = 2.47\\,$\\pm$\\,0.57\\,kpc, respectively.\n\n\n\\subsubsection{Proper motions}\n\\label{subsection.compa_proper}\n\nFor the stars that are not outliers in proper motion and parallax, we obtain mean values and standard deviation for the individual components of the proper motion of $\\mu_{\\alpha}\\cos{\\delta}$ = $-0.51$ $\\pm$ 0.48\\,mas\/yr, $\\mu_{\\delta}$ = $-1.00$ $\\pm$ 0.31\\,mas\/yr. This result agrees quite well with previous results obtained in the literature by other authors and different samples of stars. For example, \\citetads{2019A&A...624A..34Z} investigated a sample of more than 2100 stars (covering a much wider range in mass than our study) located within 7.5 degrees around the $h$~and~$\\chi$~Persei double cluster. They found for each cluster $\\mu_{\\alpha}\\cos{\\delta}$ = $-0.71$ $\\pm$ 0.18\\,mas\/yr and $\\mu_{\\delta}$ = $-1.12$ $\\pm$ 0.17\\,mas\/yr, respectively. Similar results were also obtained by \\citetads{2017MNRAS.472.3887M} and \\citetads{2019ApJ...876...65L}.\n\n\n\\subsection{Completeness of the sample}\n\\label{subsection.43_compl}\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=.49\\textwidth]{figures\/Fig7.png}\n\\caption{Color-magnitude diagram (using {\\em Gaia} photometry) of stars located within 4.5~degrees from the center of the Per~OB1 association. Colored stars shows the stars in our sample, gray circles represent the remaining stars from {\\em Gaia}, green diamonds show 17 blue and red supergiants quoted in the literature for which we lack spectra (see the last part of Table~\\ref{table.A1}). Two isochrones and a reddening vector are also included for reference purposes. See Sect.~\\ref{subsection.43_compl} for explanation.}\n\\label{figure.fig7}\n\\end{figure}\n\nAs indicated in Sect.~\\ref{subsection.21_sample}, our sample of 88 stars includes almost all blue and red Sgs (LC I and II) quoted in \\citetads{1978ApJS...38..309H}, \\citetads{1992A&AS...94..211G}, \\citetads{2010ApJS..186..191C}, and \\citetads{2014ApJ...788...58G}, plus a few LC III objects (Gs) with late-O and early-B spectral types. In particular, from a total of 107 targets quoted in these four papers that meet our selection criteria, we only lack spectra for 12 B and 5 M Sgs.\n\nTo further evaluate the completeness of our sample, we benefit from photometry provided by {\\em Gaia} and the results about parallaxes and proper motions described in Sect.~\\ref{subsection.41_plx&pm}. To this aim, we retrieved all the stars in the {\\em Gaia} DR2 catalog with $G_{mag}$ brighter than 10.5 whose parallaxes and total proper motions lie within 2$\\sigma$ of the distributions depicted in Fig.~\\ref{figure.fig5}. We then removed all stars with RUWE larger than 1.4, and those classified by the SIMBAD Astronomical Database\\footnote{\\href{SIMBAD}{http:\/\/simbad.u-strasbg.fr\/simbad\/}} as dwarfs or subgiants (luminosity classes V and IV). \n\nThe results are presented in a color-magnitude diagram (CMD) in Fig.~\\ref{figure.fig7}, where we use the same color-code as in previous figures for the stars in our sample, but this time, we also highlight the 17 stars that are classified as LC III or II-III stars in light green. \nFor reference purposes, we also include a $A_{\\rm v}$ = 1.7\\,mag reddening vector and two reddened 14\\,Myr isochrones\\footnote{Downloaded from the {\\em Mesa Iscochrones and Stellar Tracks} interface, MIST (\\citeads{2016ApJS..222....8D,2016ApJ...823..102C}).} (solid lines) shifted to a distance of 2.5\\,kpc (or, equivalently, a distance modulus of 12\\,mag.). The values of reddening for the isochrones (A$_{\\rm v}$ = 1.0 and 2.7, respectively) were selected to embrace the main-sequence band, corresponding to the region of the CMD with higher density of gray points in the bottom left corner. \n\nFrom inspection of this figure we can conclude that the level of completeness in our sample is very high, specially when we concentrate on the region of the CMD where the blue and red Sgs are located (purple, dark blue, cyan, and red stars). Interestingly, we also find that a high percentage of the 12 B~Sgs quoted in \\citetads{1978ApJS...38..309H}, \\citetads{1992A&AS...94..211G}, and \\citetads{2010ApJS..186..191C} are likely B~Gs, instead of B~Sgs. These refer to all green diamonds with $G_{mag}$ < 9, most of them classified as B Sgs in \\citetads{2010ApJS..186..191C} (see the last rows of Table~\\ref{table.A1}).\n\nIn addition, Fig.~\\ref{figure.fig7} allows us to conclude that the blue and red Sg population of Per~OB1 is affected by a variable reddening that ranges from $A_{\\rm v}$~$\\sim$~1.0 to 2.7\\,mag (in agreement with previous findings by \\citeads{2019ApJ...876...65L}), and that the age associated with the blue and red Sg population is not compatible because the higher mass present in the 14~Myr isochrone is $\\sim$14\\,$M_{\\odot}$, while all O, B, and A Gs\/Sgs included in our sample are expected to have masses higher than 20\\,$M_{\\odot}$. This latter result will be further investigated in the next paper of this series, after information about the stellar parameters of the full working sample is included.\n\n\n\\subsection{Radial velocities}\n\\label{subsection.44_rv}\n\nBy following the strategy described in Sect.~\\ref{section.3_rv}, we obtained RV estimates for all the available spectra in our sample of stars. These measurements are used (1) to investigate the RV distributions resulting from the best S\/N spectra, (2) to provide empirical constraints on intrinsic spectroscopic variability typically associated with the various types of stars, and (3) to identify spectroscopic binaries and runaway candidates.\n\n\n\\subsubsection{Best S\/N spectra}\n\\label{subsubsection.441_bestsnr}\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=.4\\textwidth]{figures\/Fig8.png}\n\\caption{RV distributions associated with the different SpT groups resulting from the analysis of the best S\/N spectra. The orange bin in the second panel from the bottom is HD\\,12842, the F Sg.}\n\\label{figure.fig8}\n\\end{figure}\n\nThe bottom panel of Fig.~\\ref{figure.fig4} shows the RVs of all stars in the sample, obtained from the best S\/N spectra, as a function of the position of the stars in right ascension. The associated distributions, this time separated by SpTs, are depicted in the form of histograms in Fig.~\\ref{figure.fig8}, with the mean and standard deviation associated with each RV distribution (after performing an iterative 2$\\sigma$ clipping) indicated at the top of the various panels. The corresponding outliers in each distribution are indicated as open squares in the bottom panel of Fig.~\\ref{figure.fig4} and listed in the fourth column of Table~\\ref{table.A4}.\n\nFrom a visual inspection of Fig.~\\ref{figure.fig8} we can conclude that except for the case of O-type stars, which has a flatter and more scattered distribution, the other three distributions are quite similar (when the outliers are eliminated), following a more or less clear Gaussian shape. (For the A\/F-type stars, only the two situated on the right-most side of Fig.~\\ref{figure.fig8} are outliers. The consequence of having fewer stars than for the B- and K\/M-type stars results in a poorer Gaussian shape.) The mean values of these three distributions are compatible within the uncertainties, with a difference smaller than 2~\\kms. Interestingly, the standard deviation of the distributions significantly drops from O- to B- and A-type stars, and continues to decrease to the K\/M-type stars (see further notes in Sects.~\\ref{subsubsection.442_multi} and \\ref{subsubsection.443_binaries}).\n\n\n\\subsubsection{Multi-epoch spectra: intrinsic variability}\n\\label{subsubsection.442_multi}\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=.49\\textwidth]{figures\/Fig9.png}\n\\caption{Measured RVs (subtracted from their mean) for a sample of 15 stars (ordered by SpT) for which we have five or more spectra, and whose detected variability in RV is more likely produced by intrinsic variability than by the orbital motion in a binary system.}\n\\label{figure.fig9}\n\\end{figure}\n\nAs indicated in the bottom panel of Fig.~\\ref{figure.fig2}, we have more than one spectrum for 73 of the stars in the sample. These observations can be used to identify binaries; however, as extensively discussed in \\cite{SimonDiaz2020}, in prep.), the effect of intrinsic variability also needs to be taken into account to minimize the spurious detection of single-line spectroscopic binaries (SB1) in the blue supergiant domain (see also further notes regarding the red supergiant domain in \\citeads{2019A&A...624A.129P,2020A&A...635A..29P}).\n\nSome examples of the type of spectroscopic variability phenomena produced by stellar oscillations or the effect of a variable stellar wind in the OBA Sg domain can be found in \\citetads{1996ApJS..103..475F, 2004A&A...418..727P, 2006A&A...457..987P, 2006A&A...447..325K, 2015A&A...581A..75K, 2017A&A...597A..22S,2018A&A...612A..40S, 2017A&A...602A..32A, 2018MNRAS.476.1234A}, for example. This effect is also illustrated in Fig.~\\ref{figure.fig9} using a subsample of 15 stars in PerOB1 for which five or more spectra are available, and whose detected variability in RV is more likely produced by intrinsic variability than by the orbital motion in a binary system (see Table~\\ref{table.A2} and further notes in Sect.~\\ref{subsubsection.443_binaries}). \n\nThese results warn us about the dangers of using a single snapshot observation to associate the outliers detected in the RV distributions shown in Fig.~\\ref{figure.fig8} with potential runaway stars or spectroscopic binaries. Some of these cases might even correspond to a single measurement in a specific phase of the intrinsic variability of the star instead of being associated with the orbital motion in a binary system or with a single star with an anomalous RV due to an ejection event. They also partially explain why the standard deviation of the RV distributions presented in Fig.~\\ref{figure.fig8} becomes smaller when moving from the blue to the red Sgs. This is just a consequence of the behavior of the characteristic amplitude of spectroscopic variability with SpT (see Table~\\ref{table.variab} and \\cite{SimonDiaz2020}, in prep.). Last, it also affects the fraction of detected SB1 stars using multi-epoch observations, or the final sample of outliers in RV (see further notes in Sects.~\\ref{subsubsection.443_binaries} and \\ref{subsubsection.523_rw&bin}, respectively).\n\nTo evaluate the effect that including information about multi-epoch spectroscopy has on the identification of outliers in the RV distribution, we have repeated the same exercise as in the case of the single-snapshot observations (Sect.~\\ref{subsubsection.441_bestsnr}), but modifying the individual measurements (obtained from the analysis of the best S\/N spectra) of stars for which four or more spectra are available by the mean of the multi-epoch RV measurements. Results of this exercise are presented in the \"RV multi\" column of Table~\\ref{table.A4}. Although the number of stars with a modified outlier status in RV is small in this specific example (only HD\\,13402 and HD\\,12953), the results presented in Fig.~\\ref{figure.fig9} indicate that it could have been larger if other epochs of the time series had been selected as single-snapshot observations.\n\n\\begin{table}\n  \\centering\n  \\caption[]{Summary of detected variability (mean and maximum of peak-to-peak amplitude of RV in each SpT group) for the sample of 15 stars depicted in Fig.~\\ref{figure.fig9}. RVs in \\kms.} \n    \\label{table.variab}\n      \\begin{tabular}{lcccc}\n\\hline\n\\hline\n\\noalign{\\smallskip}\nSpT group & N$_{\\rm stars}$ & $\\overline{\\rm N}_{\\rm spectra}$ & $\\overline{RV}_{\\rm PP}$ & RV$_{\\rm PP, max}$ \\\\ %\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\nO-type   &  2 &  6 & 21.3 $\\pm$ 0.4  & 21.7 \\\\ \nB-type   &  9 & 13 &  8.8 $\\pm$ 3.5  & 16.1 \\\\  %\nA\/F-type &  2 & 22 &  7.9 $\\pm$ 2.5  & 10.4 \\\\  %\nK\/M-type &  2 & 16 &  1.6 $\\pm$ 0.4  &  2.0 \\\\  %\n\\noalign{\\smallskip}\n\\hline\n      \\end{tabular}\n\\end{table}\n\n\n\\subsubsection{Multi-epoch spectra: spectroscopic binaries}\n\\label{subsubsection.443_binaries}\n\nGiven the stability of the FIES and HERMES instruments, and the accuracy reached in the RV measurements for most of the stars in the sample with multi-epoch spectroscopy, it might be tempted to assign the SB1 status to all stars showing a RV$_{\\rm PP}$ above a few \\kms. However, as indicated in Sect.~\\ref{subsubsection.442_multi}, the intrinsic variability in single blue supergiants can reach amplitudes of a few dozen \\kms; hence, many of these identification may lead to spurious results.   \n\nTo avoid this situation as much as possible, and in order to identify the most secure candidates to be SB1, we performed a careful inspection of the type of line-profile variability detected in each of the stars with more than one spectrum. To this aim, we mainly considered the following diagnostic lines, whenever available: \\ioni{He}{i}~$\\lambda$5875, \\ioni{Si}{iii}~$\\lambda$4552, \\ioni{O}{iii}~$\\lambda$5592, \\ioni{C}{ii}~$\\lambda$4267, \\ioni{and Mg}{ii}~$\\lambda$4481. For the case of the two red supergiants with multi-epoch observations, we found that the measured RV$_{\\rm PP}$ is lower than 2\\,\\kms, which we directly attribute to intrinsic variability.\n\nThe list of clearly identified SB1 is presented in Tables~\\ref{table.A3}. In addition to the four SB1 stars quoted there, we found five SB2 systems (some of them directly detected from a single-snapshot observation) and labeled \"LPV\/SB1?\" another five cases in which we are not entirely sure if the detected variability is due to binarity or intrinsic variability. All this information is also added to \\ref{table.A4} (column \"Spec. variability\"). \n\nWe also performed a bibliographic search for previously identified binaries in our sample of blue and red supergiants. We mainly concentrated in the works by \\citetads{2018A&A...613A..65H, 2020arXiv200505446H} and \\citetads{2019A&A...626A..20M} for the case of O-type stars, and \\citetads{1973ApJ...184..167A,1985AbaOB..58..313Z}, and \\citetads{2017A&A...598A.108L} for the B supergiant sample. In addition, we made use of \\textit{The International Variable Star Index (VSI)}\\footnote{\\href{VSI}{https:\/\/www.aavso.org\/vsx\/index.php}}. \n\nIn total, we found that six out of our sample of ten detected SB1 or SB2 systems from this work were previously identified in any of these references as such (HD\\,16429 is actually a triple system \\citepads{2003ApJ...595.1124M}). This implies four newly detected binaries: HD\\,13969, HD\\,14476, HD\\,17378 (all SB1), and HD\\,13402 (SB2).\nWe also found three binaries in the literature that were not detected from our available spectra because of short time-coverage: BD\\,+56578, an eclipsing binary \\citepads{2016AstL...42..674T, 2017A&A...598A.108L}, plus HD\\,17603 and HD\\,14956, identified as SB1 by \\citetads{2018A&A...613A..65H} and \\citetads{1973ApJ...184..167A}, respectively. All of them are labeled \"(lit.)\" in the corresponding column of Table~\\ref{table.A4}.\n\nThe stars classified as \"LPV\/SB1?\" are HD\\,13036, HD\\,13854, HD\\,13267, HD\\,14542, HD\\,12953, and HD\\,17378.\nHD\\,13036 (B0.2~III), HD\\,13267 (B6~Iab) and HD\\,14542 (B8~Iab) have RV$_{\\rm PP}$ = 10 -- 14\\,\\kms; although this value is at the boundary of the expected variability due to pulsations, which may indicate an SB1 classification, we cannot conclude after visual inspection of their line-profile variability. HD\\,13854 (B1~Ia-Iab), mostly looks like a pulsational variable, but we do not discard the possibility entirely that  this star might be a SB1. We note that \\citetads{1973ApJ...184..167A} provide RV$_{\\rm PP}$ = 24.8\\,\\kms; however, they did not consider it as a binary. For HD\\,12953 (A1~Iae), we measured RV$_{\\rm PP}$ = 10.4\\,\\kms, the largest variability in the A supergiant sample; however, after visual inspection of its line profile variability, we cannot conclude whether this is a SB1 system. \\citetads{1973ApJ...184..167A} found RV$_{\\rm PP}$ = 15.8\\,\\kms for this star, which would favor that it is an SB1. HD\\,17378 (A6~Ia) has RV$_{\\rm PP}$ = 8\\,\\kms, which is large enough to consider it as potential binary. However, we only have three spectra. \n\nLast, we found that although HD\\,14956 (B2~Ia) was classified as an SB1 with a period of $P$ = 175\\,days, and ${RV}_{\\rm PP}$ = 27.0\\,\\kms \\citepads{1973ApJ...184..167A} , and \\citetads{2017A&A...598A.108L} classified this star as $\\alpha$\\,Cygni variable, we do not see such signs of SB1 variations, as we measure RV$_{\\rm PP}$ = 5.5\\,\\kms. However, we do not have enough spectra (three) to discard this possibility.\n\nThese results about detected spectroscopic binaries, along with the RV distributions obtained from the analysis of the best S\/N spectra (i.e., obtained from a single-snapshot observation), allow us to evaluate the extent to which these distributions can be used to identify spectroscopic binaries among the outliers. We find that only four out of all the SB1\/SB2 systems detected by means of multi-epoch spectroscopy are outliers in the abovementioned distributions. In addition, some outliers have not been detected as spectroscopic binaries although more than four spectra are available for them (e.g., HD\\,13268, O8.5~IIIn, RV = $-106.2$\\,\\kms). These results can be explained when we take into account (1) that the best S\/N of some of the spectroscopic binaries correspond to an orbital phase in which the RV is close to the systemic velocity, and (2) that some outliers in RV might be runaways and not necessarily binaries. The latter situation is the case of HD\\,13268, a well-known runaway star (see also Sect.~\\ref{subsection.51_memb}). This means that if a given star is an outlier in RV, it is useful to first investigate its runaway nature (by means of its proper motion) before marking it as a potential spectroscopic binary, and vice versa; for example, although the measured RV of the B1~Ib-II star HD\\,14052 ($-90.8$\\,\\kms) deviates more than 3$\\sigma$ from the mean, this star is not an outlier in proper motion, and so we may conclude that it is more likely a spectroscopic binary than a runaway. This is confirmed through access to multi-epoch spectroscopy. We further discuss the percentage of spectroscopic binaries in our sample of stars in Sect.~\\ref{subsubsection.523_rw&bin}.\n\n\n\n\n\\section{Discussion}\n\\label{section.5_discus}\n\nTable~\\ref{table.A4} compiles and summarizes some information of interest for the discussion about membership and final identification of spectroscopic binaries and runaway stars. Columns \"$\\varpi$\" and \"$\\mu$\" indicate if a given star is part of the bulk distribution of parallaxes and proper motions, respectively, or if it is detected as an outlier of these distributions (Sect.~\\ref{subsection.41_plx&pm}). Columns \"RV best\" to \"RV final\" provide similar information for the case of RV estimates obtained from the best S\/N spectra (Sect.~\\ref{subsubsection.441_bestsnr}) for stars with four or more spectra (Sect.~\\ref{subsubsection.442_multi}), or the final distribution of RVs (Sect.~\\ref{subsubsection.521_pm}), respectively. In all these cases, different symbols are used to identify secure or doubtful cases.\n\nFor completeness, we also add to Table~\\ref{table.A4} information about confirmed spectroscopic binaries, our final decision on cluster membership status (columns \"Spec. variability\" and \"Member\"), as well as some other comments of interest for the final interpretation of results (column \"Comments\").\n\n\n\\subsection{Cluster membership}\n\\label{subsection.51_memb}\n\nAs discussed in Sect.~\\ref{subsection.41_plx&pm}, most of the stars in the sample with good astrometry (81 stars) are grouped together in the proper motion versus parallax diagram. The mean and standard deviation of the distribution of these two quantities are $\\varpi$ = 0.40 $\\pm$ 0.07\\,mas, and $\\mu$ = 1.22 $\\pm$ 0.26\\,mas\\,yr$^{-1}$ , respectively. All the stars that are located within the 2$\\sigma$ boundaries of the distribution (64 in total) are directly considered as members and labeled with filled circles in columns \"$\\varpi$\" and \"$\\mu$\" of Table~\\ref{table.A4}. The remaining 17 stars are marked with an open circle or a cross in Table~\\ref{table.A4} depending on whether they deviate by 2\\,--\\,3$\\sigma$ or more than 3$\\sigma$, respectively. We note that in this case, columns \"$\\varpi$\" and \"$\\mu$\" include information about the remaining seven stars that were not included in Fig.~\\ref{figure.fig5}: those labeled \"unreliable astrometry\" (or RUWE > 1.4). Because the information about parallaxes and proper motions is uncertain for them, we exclude these stars for the moment and mark them using brackets surrounding the corresponding symbols in columns \"$\\varpi$\" and \"$\\mu$\" of Table~\\ref{table.A4}.\n\nThese are not the only stars with unreliable parallaxes. Figs.~\\ref{figure.fig3} and \\ref{figure.fig5} include a small sample of 6 K\/M-type supergiants that despite a RUWE value well below 1.4 have larger errors than the rest of stars in the sample, and interestingly, all of them are systematically shifted to larger parallaxes (although except for one, all have total proper motions within the 2$\\sigma$ boundaries and proper motion vectors compatible with the bulk of member stars, Fig.~\\ref{figure.fig4}). They are also all marked with brackets in Table~\\ref{table.A4}.\n\nThis is likely connected to an already known problem that affects the reliability of the {\\em Gaia} DR2 astrometric solution. In brief, as pointed out by \\citetads{2011A&A...532A..13P} and \\citetads{2018A&A...617L...1C}, the position of the centroid changes on timescales of several months or a few years because of the large size and strong intrinsic photocentric variability of red supergiants. This effect leads to unreliable parallaxes and errors. \n\nA particular example of interest regarding this issue with the astrometric solution of {\\em Gaia} for the case of red supergiants is the the highly variable star HD~14528 (S~Per, $\\varpi_{Gaia}$\\,=\\,0.25\\,$\\pm$ 0.12\\,mas, $\\mu_{total\\,Gaia}$ = 2.57 $\\pm$ 0.31\\,mas\/yr). This star has an average angular size of 6.6\\,mas (\\citeads{2012A&A...546A..16R}). It was monitored for six years by \\citetads{2010ApJ...721..267A} with VLBI. The authors obtained an independent parallax of 0.413$\\pm$0.017~mas, which is just at the center of the distribution. We therefore cannot discard completely  that these six K\/M-type supergiants, which are outliers in parallax using data from {\\em Gaia} DR2, are members\nof Per~OB1.\n\nThe last star that we place in brackets is the A-type supergiant HD\\,14489. This is the brightest stars in our sample, with $G_{mag}$ = 5.1. As shown in Fig.~\\ref{figure.fig3}, this star also has much larger errors in parallax and proper motions than the bulk of stars in the sample. This may be related to the current limitation of {\\em Gaia} DR2 regarding the reliability of the astrometric solutions for stars brighter than $G_{mag} \\lesssim$ 6 \\citepads{2018A&A...616A...2L}. Another three stars share this issue, but their associated astrometric errors are much smaller and their magnitudes are close to $G_{mag}$ = 6; therefore we decided to consider their astrometric solutions reliable.\n\nTaking all this information into account, we decided to following strategy below to evaluate the membership to Per~OB1 of each star in our sample. Stars with reliable values on parallax and proper motion (i.e., not marked with parentheses in columns 2 and 3 of Table~\\ref{table.A4}) are considered as {\\em \\textup{confirmed members}} if they do not deviate more than 2$\\sigma$ from the mean of the distribution of parallaxes. For stars with unreliable values of parallax and proper motion (i.e., highlighted with brackets in columns \"$\\varpi$\" and \"$\\mu$\" of Table~\\ref{table.A4}), we adopted the following: if they are not outliers in parallax, they are considered {\\em \\textup{likely members}}; if they are outliers in parallax, we consider them {\\em \\textup{candidate members}}, except for the K\/M-type stars, which remain likely members because of the arguments provided above. Last, stars with reliable astrometry that are outliers in parallax (as well as those stars in IC\\,1805, see below) are considered {\\em \\textup{nonmembers}}.\n\nMost of the stars are properly classified using these criteria. However, a few cases deserve further attention.\n\\paragraph{HD~13022 (O9.7~III) and HD~12842 (F3~Ib):} These two stars are classified as members following the guidelines above, but they are outliers in proper motion (Fig~\\ref{figure.fig5}). Interestingly, they have a very small proper motion compared to the rest of the stars in the sample (see in Fig.~\\ref{figure.fig4} the two stars with very small vectors located at (RA, DEC)\\,$\\sim$\\,(32, 58.5)\\,deg). Awaiting a more detailed study of these two stars, we continue considering them members for the moment.\n\\paragraph{HD~16691 (O4~If), HD~15642 (O9.5~II-IIIn), HD~13745 (O9.7~II(n)), and HD\\,13268 (ON8.5~IIIn):} These four O-type stars are clear outliers in proper motion (see Figs.~\\ref{figure.fig5} and \\ref{figure.fig4}). We consider the first three runaway members  because their parallaxes lie within the 2$\\sigma$ boundaries. The fourth (HD~13268) is an interesting case; although this star has a somewhat larger parallax, it has a RV of $\\sim$ 105\\,\\kms. Therefore, given its spectral classification and this high RV pointing to us, it can still be considered a runaway member of Per~OB1. This star is a well-known fast-rotating nitrogen-rich O-type runaway star (e.g., \\citeads{1972AJ.....77..138A}; \\citeads{1989A&AS...81..237M}; \\citeads{2014A&A...562A.135S}; \\citeads{2015A&A...578A.109M}; \\citeads{2017A&A...603A..56C,2017A&A...604A.123C})\n\\paragraph{HD~14322 (B8~Iab):} This star is an outlier in parallax with a value of $\\varpi$ = 0.21$\\pm$0.04\\,mas. Although the TGAS catalog provides a value for it of $\\varpi$ = 0.44$\\pm$0.38\\,mas (within the boundaries of $\\varpi$), the error is much larger. This inconsistency caused us to modify its status from nonmember to member candidate while awaiting {\\em Gaia} DR3.\n\n\\paragraph{HD~14489 (A1~Ia):} This is a bright A-type star ($G_{mag}$ = 5.1), outlier in parallax, and with the largest parallax error. Although it has a RUWE = 0.81, we do not trust its {\\em Gaia} astrometry, as explained before, because of its brightness. The result from TGAS provides a parallax of $\\varpi$ = 0.45 $\\pm $0.94\\,mas, and although it is within the adopted boundaries of Per~OB1, the error is very large. This star is also an outlier in RV and close to the 2$\\sigma$ boundary in proper motion. Therefore we decide to label it a runaway member candidate.\n\n\\paragraph{BD+56724 (M4-M5~Ia-Iab):} This star has the largest parallax in Fig.~\\ref{figure.fig5}, and a RUWE = 0.93. Although the reliability of {\\em Gaia} DR2 parallaxes for the K\/M-type stars may be low, its large deviation from the mean of the distribution could mean that this star is not a member. It is also an outlier in proper motion, but its magnitude and RV are similar to other red supergiants in the sample. We therefore retain this star as member candidate for the moment.\n\n\\paragraph{HD\\,15570 (O4~If), HD15558 (O4.5~III(f)), HD16429 (O9~II(n)), and BD~+60493 (B0.5~Ia):} All these stars are located within or in the surroundings of IC\\,1805. Interestingly, all of them but one are O-type stars. Although they are located within the 2$\\sigma$ boundaries of the parallax and proper motion distribution (except for HD~16429, but this is a triple system with a RUWE = 8.8), we decided to mark them nonmembers based on their separated location in the sky and their direct connection with the surrounding H~{\\sc ii} region. They seem to be linked to a younger star-forming region located at higher galactic latitudes (but at the same distance). Most of them are also outliers in RV (see Table~\\ref{table.A4}), but this is likely due to their binary nature. \n\nThe final result of this classification, also taking into account the comments on some individual stars presented above, is summarized in column \"Member\" of Table~\\ref{table.A4}. In total, we have 70 confirmed members, 9 likely members, 5 member candidates, and 4 nonmembers. Interestingly, only stars in IC\\,1805 are finally classified nonmembers. The remaining 84 stars likely belong to the Per~OB1 association (although some of them are identified as runways, see Sect.~\\ref{subsubsection.523_rw&bin}).\n\n\n\\subsection{Kinematics.}\n\\label{subsection.52_kinem}\n\nIn Sect.~\\ref{subsection.41_plx&pm} and \\ref{subsection.44_rv} we provided a global overview of the results about proper motions and RVs for the complete sample of stars, also including some information about identified spectroscopic binaries. In this section we discuss these results more in detail. We also refer to  \\citetads{2017MNRAS.472.3887M, 2019A&A...624A..34Z, 2020MNRAS.493.2339M} for complementary (and in some cases more detailed) information about the global and internal kinematical properties of stars in the Per~OB1 association.\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=.95\\textwidth]{figures\/Fig10.png}\n\\caption{Radial distance from the center of Per~OB1 against the mean RV for the stars with more than one spectrum, or RV for the stars with only one spectrum, excluding the SB2 binaries. The stars within 1\\,degree from IC\\,1805 were also excluded. The filled gray rectangle shows 2$\\sigma$ of all the RVs, excluding those from stars identified as binaries (SB1 or SB2). The filled colored circles denote stars identified as SB1. The filled colored crosses denote starts outside 2$\\sigma$ from the mean that are not identified as binaries. The colored stars show the remaining stars. The colored error bars present RV$_{\\rm PP}$ for stars with multi-epoch data, except for SB1. The open triangles show stars that are outliers in proper motion and were therefore identified as runaways.} \n\\label{figure.fig10}\n\\end{figure*}\n\n\n\\subsubsection{Proper motions}\n\\label{subsubsection.521_pm}\n\nFigure~\\ref{figure.fig5} and the top panel of Fig.~\\ref{figure.fig4} provide a global overview of the distribution of proper motions in the whole star sample. These figures show that (except for a few outliers) most of the stars in our sample that are located below the Galactic plane (among them, those in $h$~and~$\\chi$ Persei) can be considered a dynamically connected population of stars. This result perfectly agrees with previous findings by \\citetads{2008ApJ...679.1352L}. Using proper motions from the {\\em Hipparcos} mission, these authors showed that the luminous members of the Per~OB1 association exhibit a bulk motion away from the Galactic plane, such that their average velocity increases with height above the Galactic plane.\n\nFurthermore, inspection of the results for proper motions (relative to h\\,and\\,$\\chi$~Persei) and parallaxes presented in Fig.~\\ref{figure.fig6} and Table~\\ref{table.radiplxpm} allows us to conclude that the distributions of parallaxes associated with stars located at increasing distances from the center of the double cluster are all compatible (at least we find no clear subgrouping in terms of parallax at least given the accuracy of {\\em Gaia} DR2 astrometry -- except maybe the stars in $h$~and~$\\chi$~Persei because their parallax is  somewhat larger or their distance is somewhat closer). We also conclude that the mean and standard deviation of the distribution of proper motions in $h$~and~$\\chi$~Persei is much smaller than in the more extended population. \n\nMoreover, the spatial distribution of proper motions in the extended population of blue and red supergiants in Per~OB1 does not follow an expanding structure centered in the $h$~and~$\\chi$~Persei double cluster. Instead, the local proper motions of most of the stars located north of these clusters seem to point outwards from an imagined center located at about 1 degree north of the double cluster (see also \\citeads{2018ARep...62..998M}, \\citeads{2020MNRAS.493.2339M}). These results from the proper motion, linked with the results by \\citetads{2008ApJ...679.1352L} mentioned above, are compatible with a scenario in which the halo population of blue and red supergiants around the double cluster has been formed from a more diffuse region of interstellar material compared to the denser region associated with the clusters themselves. \n\nIn addition, four O-type stars south of the region can be clearly considered runaways based on the size and direction of their proper motions (see also Fig.~\\ref{figure.fig5}). Interestingly, their proper motion vectors do not point outward from $h$~and~$\\chi$~Persei, but to a far more extended region of the Galactic plane (see also the discussion in Sect.~\\ref{subsubsection.523_rw&bin}). Finally, as also indicated in Sect.~\\ref{subsection.51_memb}, the stars located within or near IC~1805 likely belong to a younger population of stars that is not necessarily connected with the remaining stars in Per~OB1.\n\n\n\\subsubsection{Radial velocities}\n\\label{subsubsection.522_rv}\n\nFigure~\\ref{figure.fig8} and the bottom panel of Fig.~\\ref{figure.fig4} summarize the RV results obtained with the best S\/N spectra. The analysis of these spectra has allowed us to characterize the RV distributions for the different SpT groups and to identify potential spectroscopic binary systems and runaway stars among the outliers of the distributions (see columns \"RV best\" and \"RV multi\" in Table~\\ref{table.A4}). We then illustrated in Sect.~\\ref{subsubsection.442_multi} (see also Fig.~\\ref{figure.fig9}) the importance of incorporating information from the analysis of multi-epoch observations for the correct interpretation of the RV distribution, and in particular, to avoid the spurious identification of spectroscopic binaries (either from a single epoch or from multi-epoch observations) due to the effect on the measured RVs of the intrinsic variability caused by stellar oscillation and\/or wind variability in the blue supergiant domain.\n\nLast, we learned that after eliminating outliers associated with confirmed spectroscopic binaries (via multi-epoch spectroscopy) and runways (via proper motions), the RV distributions for the B, A\/F, and K\/M~Sgs are fairly compatible in terms of mean values and standard deviations. In addition, we found that most of the O-type stars in the sample are either (1) runaways, as detected from the proper motions, (2) spectroscopic binaries, or (3) are considered nonmembers because they are located nearby IC\\,1805, far away from the main distribution of stars in Per~OB1. As a result, the RV distribution of the O-type sample is remarkably broader than for those associated with the other SpT.\n\nWe now take all these results into account to provide final information about RVs in Fig.~\\ref{figure.fig10} and in the column \"RV final\" of Table~\\ref{table.A4}. To do this, we first replaced the list of measurements obtained from the best S\/N spectra by the mean value resulting from the analysis of the multi-epoch observations for those cases for which we have more than one spectrum. Then we used this list of values, except for all the SB2 binaries, and the stars identified as nonmembers (see Sect.~\\ref{subsection.51_memb}), to obtain the mean and standard deviation by performing an iterative 2$\\sigma$ clipping. \n\n\\begin{table*}\n  \\centering\n  \\caption[]{Summary of the number of outliers in proper motion and RV that are used for the final identification of runaway stars. In the case of the proper motion, we indicate cases that deviate by more than 2$\\sigma$ from the mean of the distribution for each individual component and the total proper motion. In the case of RV, we separate cases that deviate by more than 2$\\sigma$ and 4$\\sigma$, respectively. In parentheses, we indicate targets whose outlier characteristic is not entirely clear from the available data. The last column indicates the final number and percentage of clearly detected runaways for each SpT group.}\n    \\label{table.rw}\n      \\begin{tabular}{lccccccccc}\n\\hline\n\\hline\n\\noalign{\\smallskip}\nSpT & \\multicolumn{3}{c}{PM} & & \\multicolumn{2}{c}{RV} & & \\multicolumn{2}{c}{Runaways} \\\\ %\n\\cline{2-4} \\cline{6-7} \\cline{9-10}\n\\noalign{\\smallskip}\n      &  $\\mu_{\\alpha}\\cos{\\delta}$      & $\\mu_{\\delta}$   & $\\mu_{\\rm Total}$ &  &   $>$2$\\sigma$ &   $>$4$\\sigma$ &  & \\# & \\% \\\\ %\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\nO-type   &  3  &  5    & 5    & & 7~(+1) & 3 & & 5 & 45 \\\\\nB-type   &  1  &  2    & 2    & & 9~(+6) & 1 & & 2 &  5 \\\\ %\nA\/F-type &  1  &  1(+1)& 1(+1)& & 3 & 1 & & 1(+1) &  1(+1) \\\\ %\nK\/M-type & (1) & (1)   & (1)  & & 2 & 0 & & (1) &  (5) \\\\ %\n\\noalign{\\smallskip}\n\\hline\n      \\end{tabular}\n\\end{table*}\n\nThe results of this process are presented in Fig.~\\ref{figure.fig10}, where the RVs of all stars that were not excluded from the list are presented against the radial distance from the center of Per~OB1. The obtained mean and standard deviation are shown in the top right corner ($-42.9$ $\\pm$ 3.5\\,\\kms) of the figure, and the horizontal gray band indicates the 2$\\sigma$ boundaries. \n\nAlthough most of the stars in the sample are concentrated within the central 100\\,pc, we observe that except for a few cases, the remaining stars also lie within the 2$\\sigma$ boundaries. Therefore, once more, and as was suggested by \\citetads{2010ApJS..186..191C} and \\citetads{2019A&A...624A..34Z}, the extended population of blue and red Sgs in Per~OB1 (up to 200~pc, i.e., relatively far away from $h$~and~$\\chi$~Persei) seems to have a common origin in terms of kinematics. No global gradient (as a function of distance to the center of the association) or local substructures are observed in the distribution of RVs. \n\nAs indicated above, we obtain $\\overline{RV}$ = $-42.9$ $\\pm$ 3.5\\,\\kms\\ using the whole sample of stars that are not excluded from the list. Regarding $h$~and~$\\chi$~Persei, we obtained average values of RV$_{\\rm \\chi\\,Per}$ = $-44.4$ $\\pm$ 1.4\\,\\kms and RV$_{\\rm h\\,Per}$ = $-41.1$ $\\pm$ 2.6\\,\\kms, respectively. These results agree well with those previously obtained by other authors. For the association as a whole, \\citetads{2017MNRAS.472.3887M} provided a mean value of $\\overline{RV}$ = $-43.2$ $\\pm$ 7.0\\,\\kms\\ using available information of member stars from the TGAS catalog. For the individual clusters, \\citetads{1991AJ....102.1103L} provided RV$_{\\rm \\chi\\,Per}$ = $-44.4$ $\\pm$ 0.7\\,\\kms and RV$_{\\rm h\\,Per}$ = $-46.8$ $\\pm$ 1.7\\,\\kms, respectively, using a sample of cluster stars (mainly early type, more specifically, B- and A-type stars). In the particular case of $h$\\,Per, the fact that we have only three suitable stars to compute the mean may explain the poorer agreement. \n\nAs in the case of the analysis of the best S\/N spectra and the multi-epoch observations, we provide in column \"RV final\" of Table~\\ref{table.A4} a list of identifiers to separate the outliers of the distribution of final values of RV from stars within the 2$\\sigma$ boundaries. This information is used in the next section to determine additional SB1 stars that have not previously been identified based on the available multi-epoch spectra.    \n\n\n\\subsubsection{Runaway and binary stars.}\n\\label{subsubsection.523_rw&bin}\n\n\n\\begin{table*}\n  \\centering\n  \\caption[]{Summary of the number of binary stars in the sample (see Table~\\ref{table.A4}). For columns \"SB1\" and \"SB2\", the percentage shows the fraction with respect to the total number for each SpT. We split the B-type stars into two groups to separate giants from supergiants. Column \"Lit.\" counts the number of binary stars found in the literature. Column \"SB1?\" counts the sum of the stars labeled \"LPV\/SB1?\" in column \"Spec. variability\" and \"SB1?\" in column \"Comments\" (we note that if a star is labeled both as \"LPV\/SB1?\" and \"SB1?\", we only count the first). The total number of stars are in column N$_{\\rm All}$. Column \"\\% bin\" gives the percentage of total and potential binary stars with respect to the total number of stars.} \n    \\label{table.bin}\n      \\begin{tabular}{cccccccc}\n\\hline\n\\hline\n\\noalign{\\smallskip}\nSpT & SB1 & SB2 & Lit. & SB1? & N$_{\\rm All}$ & \\% bin. \\\\ %\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\nO             & 0        & 2~(15\\%)   & 1 & 1 & 13 & 15 -- 30\\% \\\\\nB~I \\& II     & 3~(8\\%)  & 1~(3\\%)    & 1 & 5 & 37 & 10 -- 27\\% \\\\ %\nA\/F          & 0         & 0          & 0 & 2 & 11 & 0 -- 18\\% \\\\ %\nK\/M          & 0         & 0          & 0 & 2 & 18 &  0 -- 10\\% \\\\ %\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\nB~III         & 2~(22\\%) & 2~(22\\%)   & 0 & 1 &  9 & 45 -- 55\\% \\\\ %\n\\noalign{\\smallskip}\n\\hline\n      \\end{tabular}\n\\end{table*}\n\n\nThe last two columns of Table~\\ref{table.rw} summarize the final number and percentage of identified runaways in each of the four SpT groups. We proceed as follows to obtain this. First, we assigned the runaway status to all outliers in proper motion (meaning that the magnitude of any of their individual components, or the total proper motion, deviates by more than 2$\\sigma$ from the mean of the corresponding distribution). Then we considered the possibility of identifying additional runaway stars through their RVs. In this case, we decided to only label them (if not detected as SB2) clear runaways if their RV deviates by more than 15~\\kms from the mean of the final RV distribution presented in Fig.~\\ref{figure.fig10}. We made this decision based on two arguments: the first refers to the result presented in the fifth column of Table~\\ref{table.rw}; namely, the number of identified runaways when all outliers in RV are considered that deviate by more than 2$\\sigma$ is too large when compared with those detected through PMs. The second argument is based on the results of RV$_{\\rm PP}$ that is expected to be produced by intrinsic variability, which can be as high as 10\\,--\\,20~\\kms\\ in some cases (see Table~\\ref{table.variab}).\n\nThese arguments are supported by the fact that it is very unlikely to find a runaway star that is an outlier in RV, but not in at least one of the components of the proper motion. In contrast, as described above, intrinsic variability can lead to single-snapshot RV measurement that can easily deviate by up to 10\\,--\\,20~\\kms (or, equivalently, about 4\\,--\\,5$\\sigma$ in this specific sample of stars). Furthermore, this situation can be even more dramatic for large-amplitude SB1 systems for which only a low number of spectra is available. It is therefore more likely that a star that is not an outlier in proper motion but is an outlier in RV is an SB1 than a runaway. Alternatively, if the deviation in RV is smaller than the typical intrinsic variability corresponding to the associated SpT, it might not even be a binary star.   \n\nA practical example of the latter situation is the BN2~II-III star BD\\,+56578, for which we only have three spectra that cover a very short time-span (one day). This star is not an outlier in proper motion, but is an outlier in RV (it deviates by 13$\\sigma$ from the mean of the RV distribution). Based on what we have described above, this star should be labeled as a potential binary, a suspicion that is confirmed from the literature (\\citeads{2017A&A...598A.108L}).\n\nFollowing these arguments, all stars in Fig.~\\ref{figure.fig10} whose RV measurements deviate by more than 2$\\sigma$ (i.e., which lie outside the gray band) and up to 10\\,--\\,20~\\kms and that have not previously been detected as runaways through proper motions (open triangles) or as spectroscopic binaries through multi-epoch spectroscopy (filled circles) are quite likely single pulsating stars.\n\nOverall, we identify a total of 11 runaway stars. The group of stars with a larger number of runaways (45\\%) are the O-type stars. This is followed by the B and K\/M Sgs, with 5\\% each. (We note, however, that the runaway status of the M Sg BD\\,+56724 can be a spurious result because it is based on the $Gaia$-DR2 proper motion, which may not be as reliable as for the other stars because of the problems regarding size and variability of the red Sgs.) Last, the lower percentage of runaways is found for the A Sgs, with only 1 or 2\\%, depending on whether we trust the $Gaia$-DR2 proper motion of the bright star HD14489, which also has a much larger parallax than the remaining stars in Per\\,OB1.\n\nIt thus becomes clear again that a high percentage of the O-type stars in the Per~OB1 region can be considered a dynamically distinct group. However, in contrast to previous assumptions (see, e.g., \\citeads{2002AJ....124..507W}), the fact that all of them are found within the 2$\\sigma$ boundaries of the parallax distribution indicates that they belong to the same grouping as the remaining blue and red supergiants in Per~OB1, and not to a more distant, dispersed association. Although further confirmation is needed, the most likely origin of the O-star runaways is a dynamical kick by a supernova explosion in a previously bounded binary system. This hypothesis is reinforced by the fact that none of the detected runaways are identified as binary systems (and the other way round).  \n\nTable~\\ref{table.bin} summarizes the results for the detected binaries, again separated by SpT group, and this time differentiating the B Sgs from the B Gs because they represent the evolutionary descendants of main-sequence stars in two different mass domains. We refer to Sect.~\\ref{subsubsection.442_multi} for a description of how the SB1 and SB2 stars where identified. The targets labeled \"SB1?\" include targets fulfilling any of the two following criteria. On the one hand, stars with five or more spectra for which we cannot clearly decide whether the detected line-profile variability is due to intrinsic variability or orbital motion. On the other hand, following the arguments above, we identified stars as \"SB1?\" whose RV$_{\\rm PP}$ is larger than the typical intrinsic variability expected for their SpT (see Fig.~\\ref{figure.fig9}).\n\nThe main conclusions from inspection of the results presented in Table~\\ref{table.bin} (and Table~\\ref{table.rw}) are summarized as follows. First, the percentage of detected spectroscopic binaries decreases toward later SpT, or equivalently as the massive star evolution proceeds. This result agrees with recent findings by \\citetads{2017IAUS..329...89B}; \\citetads{2019A&A...624A.129P,2020A&A...635A..29P}; Sim\\'on-D\\'iaz et al. (2020, subm.). When we assume that the detected runaways indicate a past binary evolution, the total percentage of clear binaries (excluding those labeled \"SB1?\") would decrease from $\\sim$60\\% to $\\sim$15\\% when the O star and B~Sg samples are compared, and further below $\\sim$5\\% when the cooler Sgs are considered. Second, while the decreasing tendency remains in both cases, the exact behavior of the percentage of detected spectroscopic binaries is different depending on whether we also include the stars labeled \"SB1?\" stars. Therefore it is critical to confirm or dismiss our suspicion that most of the stars with RV$_{\\rm PP}$ below 10-15~\\kms\\ are actually single pulsating stars and not spectroscopic binaries. Access to multi-epoch data for the whole sample of star is therefore crucial to obtain reliable empirical information about the relative percentage of binaries throughout the massive star evolution. Finally, as an aside, the percentage of spectroscopic binaries is much higher among the B Gs than in the B Sgs.\n\nThis clearly shows that any further attempt to interpret the empirical properties of this sample of massive stars in an evolutionary context must take into account that a large fraction of the O stars is or likely has been part of a binary or multiple system. In addition, some of the other more evolved targets may also have been affected by binary evolution. \n\n\n\n\n\\section{Summary and future prospects.}\n\\label{section.6_summary}\n\nOur study has provided all the necessary environmental information that will be used in a forthcoming paper, in which we will also incorporate results obtained from a quantitative spectroscopic analysis of the whole sample (including stellar parameters and surface abundances) to perform a complete homogeneous characterization of the physical and evolutionary properties of the massive star population of the Per~OB1 association.\n\nIn this paper, we have studied a sample of 88 massive stars located within 4.5\\,deg from the center of the Per~OB1 association using high-resolution multi-epoch spectroscopy, and astrometric information from the {\\em Gaia} second data release (DR2). \n\nWe have investigated membership of all star in the sample to the Per\\,OB1 association, resulting in 70 members, 9 likely members, and another 5 candidates that require further investigation, while the other 4 were considered nonmembers as they belong to IC\\,1805.\n\nWe have found eight clear and two likely runaway stars, most of them O-type stars. We also identified 5 SB1 and five SB2 stars (these include three and one new binary systems, respectively), plus another 11 potential SB1 stars that we propose are single pulsating stars. \n\nTo obtain these results, we took their parallaxes and proper motions (as compiled from {\\em Gaia} DR2) into account, and the RV estimates obtained from the available multi-epoch and\/or single snapshot spectra. In addition, we also considered the reliability of the astrometry provided by {\\em Gaia} through the RUWE value, the potential decrease in reliability of {\\em Gaia} astrometry in the case of the red Sgs because of their large size and photocentric variability, and the expected amplitude of spectroscopic variability produced by stellar pulsations and\/or wind variability when spectroscopic binaries are identified based on their RV measurements. \n\nWe have also analyzed some global properties of the sample and obtained averages in parallax, total proper motion, and RV of $\\varpi$ = 0.40 $\\pm$ 0.07\\,mas, $\\mu$ = 1.22 $\\pm$ 0.26\\,mas\\,yr$^{-1}$ ($\\mu_{\\alpha}\\cos{\\delta}$ = -0.50 $\\pm$ 0.48, $\\mu_{\\delta}$ = -0.99 $\\pm$ 0.31), and $-42.9$ $\\pm$ 3.5\\,\\kms. All these results agree relatively well with previous studies based on different stellar samples comprising the Per~OB1 association (some of them focused on the h~and~$\\chi$ Persei clusters). \n\nGenerally speaking, no important differences are detected in the distribution of parallaxes, proper motions, and RVs when stars in h~and~$\\chi$ Persei or the full sample are considered, which suggests a very extended dynamically interrelated population. However, a few clear outliers in the proper motion and RV distributions are also found. A large fraction of these are O-type stars (almost 50\\%). The further analysis of their proper motions and RVs indicates that they are runaway stars, probably resulting from the kick of a supernova explosion in a previously bounded binary system.\n\nFinally, we have found that the percentage of secure binaries decreases from the hotter to the cooler Sgs. In particular, this percentage decreases from 15\\% to 10\\% when the O star and B~Sg samples are compared (or alternatively, from 60\\% to 15\\% when we consider the runaway stars as previous binaries), and it practically vanishes in the A\/F and K\/M Sgs. Further investigation of the potential connection between this result and merging processes that occur during the evolution of massive stars is an interesting direction of future work.\n\n\n\n\\begin{acknowledgements}\n\nBased on observations made with the Nordic Optical Telescope, operated by NOTSA, and the Mercator Telescope, operated by the Flemish Community, both at the Observatorio de El Roque de los Muchachos (La Palma, Spain) of the Instituto de Astrof\\'isica de Canarias. We acknowledge funding from the Spanish Government Ministerio de Ciencia e Innovaci\u00f3n through grants PGC-2018-091\\,3741-B-C211\/C22, SEV 2015-0548, and CEX2019-000920-S and from the Canarian Agency for Research, Innovation and Information Society (ACIISI), of the Canary Islands Government, and the European  Regional Development Fund (ERDF), under grant with reference ProID2017010115. This research made use of the SIMBAD, operated at Centre de Donn\\'ees astronomiques de Strasbourg, France, and NASA's Astrophysics Data System. The background images were taken from The STScI Digitized Sky Survey (\\href{http:\/\/archive.stsci.edu\/dss\/copyright.html}{Copyright link)}.\n\n\\end{acknowledgements}\n\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nInternet of Things (IoT) is an omnipresent technological field with a focus on ease of use as well as a high degree of device autonomy. However, it presents a host of new issues regarding data security as well as substantively increasing the attack surface of the average end user's device ecosystem. The purpose of these IoT devices is to incorporate `smart' automation into the everyday life of the end user, but just as the function of these devices is automatic, the access control mechanisms are as well. As IoT devices become more prevalent in normal user environments, as well as integrated into our societal infrastructure, this automatic data sharing becomes a potential weak-point in the chain of data security. \n\nSecuring the data handled by smart autonomous devices becomes more important as the ubiquity of connected devices grows. If one home in a neighborhood has an IoT connected thermostat, there is not much incentive for adversaries to develop technologies to exploit potential underlying weaknesses. However, if every other house contains a plethora of smart devices, each of which is continuously gathering and transmitting data, there is a much higher potential gain from compromising these devices\\cite{205156}. Reality is becoming more reflective of this hypothetical every day. The amount of smart devices worldwide increased by one billion from 2019 to 2020. The global IoT market is projected to nearly triple between 2020 to 2030, from 8.74 to 25.44 billion devices\\cite{holst_2021}. \n\nWith this substantial increase in amount of connected devices, the infection rate of these devices is similarly growing. In 2019, compromised IoT devices made up 16.17\\% of all infected devices connected to mobile networks, that number more than doubled to 32.72\\% in 2020\\cite{onestore}. \nThe type of data being secured is also changing as these technologies are adopted across different fields, and therefore so are the consequences of data being compromised. Manufacturing environments become `smarter' every day as they integrate IoT devices to increase efficiency and decrease production time. In the US alone, manufacturing is a 2.3 trillion dollar industry which accounts for 11.39\\% of GDP\\cite{nam}. Interruption to these processes on a macro scale could very well lead to fiscal losses in the billions. Further, smart internet connected cars are becoming more widespread every year, with 51.1 million being sold in 2019 and a projected 76.3 million to be sold in 2023\\cite{wagner_2020}. This far-reaching growth leads to higher quality and convenient cars for consumers, however internet connected cars must have sufficient security protocols in place. The reliability and integrity of smart and autonomous car data is critical when user's lives depend on the vehicle functioning as designed. \n\nThe gravity of these security issues highlight the need for more secure frameworks and practices regarding the handling of data generated by IoT devices and connected ecosystem. One way to accomplish this is the integration of \\textbf{digital twins} \\cite{digitaltwins} into device control and data acquisition. Digital twins, or device shadows\\footnote{While digital twin refers to the whole encapsulation of a physical device in software and device shadow refers to the JSON data structure holding a representation of device state, these terms are very similar and are used interchangeably in this paper.}, are the virtual counterpart to physical objects which introduce a layer of abstraction between higher level control of devices and device specific actuation and sensing methods. These shadows can be used to facilitate separation between the object and cloud services layer (detailed in the background section), as well as enabling separation of IoT data into subsets. Digital twins also lead to more consistent interaction between higher level layers and physical devices. Device state as well as current connection status can always be accessed by higher level layers due to the persistent nature of the twin. It should be noted that digital twins comprise only a portion of the overarching architecture. The separation of device communication into layers within an access control oriented (ACO) architecture (discussed in the related work section) is necessary and present. Authentication and subsequently authorization must happen between discrete layers to ensure the architecture as a whole is secure. \n\nIoT devices often have multiple state data such as user settings, manufacturer configuration, and operational status. Each of these state data needs to be accessed by different users or at differing frequencies. Usually, there is one to one mapping between physical devices and virtual objects, meaning, one can only associate a single virtual object to a single device and are required to store all sets of device state data in one shadow. As a limiting consequence, all users will have access to the entire shadow and can consequently read and update state data they should not have. Further, data tagging plays a large role in our access control mechanisms and serves as the basis for the separation of data in the architecture. Tags are attached to the data generated in the system and are the attribute on which data is separated. This allows for easy and computationally inexpensive grouping of like data and adds further classification for data type. In this way, tags are the metric on which data is separated, and the digital twins are the receptive containers for that data. Once data has been separated and distributed based on the tags it carries, access control is centered around granting access to individual shadows. The data present in a given shadow is directly related to the tags applied to that data, therefore granting access to individual subsets of data within shadows is a form of Tag Based Access Control (TBAC). \n\nIn this paper, we propose a novel approach of data security by using multiple shadows (digital twins) for one physical object, with the intent of separating data among different virtual objects based on tags assigned on the fly, which are then used to limit access to different data points by any authorized users\/applications. The proposed solution is deployed at the edge, supporting low latency and real time security mechanisms with minimal overhead, and is light-weight as discussed in the implementation section. The implementation described in this paper is built on version 5.0 of the MQTT\\footnote{https:\/\/mqtt.org\/} protocol, and therefore communication occurs within topics in a publisher-subscriber model. While the referenced implementation applies access control to these topics, any model must include similar access control techniques with regard to flow of information between layers. For proof-of-concept, we focus on the integration of digital twins and TBAC in two industry applications: smart vehicles and smart manufacturing. We will examine the mechanisms for secure data sharing between digital twins, the advantages of tagging data in a digital twin system, and the performance impacts of the proposed data separation scheme. \n The key contributions of this paper are as follows:\n\\begin{itemize}\n\\item Attachment of tags directly to device state information in order to reduce `distance' between access control mechanisms and device data itself.\n\\item Dynamic and on-the-fly subdivision of device state at the local edge according to attached tags.\n\\item Limiting data exposure to authorized entities via subdivision of data in a many-to-one relationship between digital twins and physical devices.\n\\item Implementation of the proposed architecture to reflect the plausibility and efficiency, together with\nbrief comparative discussion on performance metrics.\n\\end{itemize}\n\n\n\n\n\nThe remainder of the paper is structured as follows. Section \\ref{sec:background} discusses relevant background such as established access control oriented (ACO) architectures and IoT literature. Section \\ref{sec:need-for-edge} demonstrates the necessity of security at the local edge in abstract principle as well as in applications such as intelligent vehicles and smart manufacturing. Existing industry solutions and their limitations are also examined in this section. Section \\ref{sec:tbac-proposal} defines the proposed architecture and the mechanisms for the attachment of tags within the context of TBAC. Section \\ref{sec:evaluation} presents implementation and associated performance metrics. Finally, section \\ref{sec:conclusion} summarizes our work and looks ahead to future work in this field. \n\n\n\\section{Relevant Background}\n\\label{sec:background}\nThis section reviews primitive building blocks of cloud and edge assisted smart connected systems. In addition, we will also reflect on relevant literature which has offered some security solutions and approaches for IoT and CPS ecosystems. \n\n\\begin{figure*}[th!]\n  \\centering\n  \\includegraphics[width=.7\\textwidth]{Diagrams\/Architectures.png}\n  \\caption{Multi-layered Access Control Focused Architectures}\n  \\label{Architecture_Layers}\n\\end{figure*}\n\\subsection{Access Control Oriented Architectures}\nSeveral access control oriented (ACO) architectures have been proposed in the literature for IoT \\cite{7809752,gupta2018authorization,Alshehri2018AccessCM,8673782,weijia2018,weijia2021,userauthIoT,celik_tan_mcdaniel_2019,franch2020,yahyazadeh2019,yahyazadeh2020,gupta2020attribute,dgupta2020access} and cyber physical systems (CPS) such as smart cars \\cite{gupta2018authorization}\\cite{guptaABAC2019}, intelligent transportation \\cite{gupta2020secure} and smart manufacturing \\cite{9502070}, which focus on the separation of systems into layers as illustrated in Figure \\ref{Architecture_Layers} (a). As shown in\nFigure \\ref{Architecture_Layers} (b), ACO architecture (proposed by Alsehri and Sandhu\\cite{7809752}) has four layers - object, virtual object, cloud services and application \u2013 with users and administrators\ninteracting at object and application layers. In addition, communication can happen\nwithin a layer (shown as self loop in Figure \\ref{Architecture_Layers} (b) and the adjacent\nlayers above and below. It should be noted that the extended access control oriented (E-ACO) \\cite{gupta2018authorization} architecture shown in Figure \\ref{Architecture_Layers} (b) is an extension to the generic ACO architecture with some additional components as discussed in the following section. \n\nThe \\textit{object layer} is comprised of the physical devices which either sense or actuate the environment within which they reside. These devices can be individual or clustered into larger objects (shown in Figure \\ref{Architecture_Layers} (b)) which contain many sensors, actuators, etc. There are several examples of clustered objects such as smart cars, mobile phones, or production lines; all of which contain many smart devices connected to a network. The physical objects in the object layer communicate with their digital twins (aka virtual objects) in the virtual object layer. These devices can communicate with other devices using different communication technologies\nincluding Bluetooth, WiFi, Zigbee, LAN, LTE or 5G. Physical devices communicate with their cyber counterparts (virtual objects) using protocols like HTTP, MQTT, DDS or CoAP. Users can also\ndirectly access physical objects at this layer. In an extended access control oriented architecture (E-ACO) as shown in Figure \\ref{Architecture_Layers} (b), clustered objects (COs) are introduced, which are objects with multiple sensors, and allow for possible interaction between sensors in same CO or between different object's sensors. These COs, such as smart cars, also have applications associated with them which offer services to users, in this case drivers. For example, a rear vision system is an application in cars to get rear-view, which gets data from the rear camera (an object) to provide dashboard view to the driver. These applications\nin the object layer of E-ACO are add-on's to the object layer in ACO architectures.\n\nThe \\textit{virtual object} layer holds the digital twins for all of the physical objects in the system. Digital twins in this layer communicate directly with their associated physical objects, the other virtual objects (VOs) present, and the cloud layer. The VOs in this layer hold the last received state of the physical object they represent, as well as processing desired states for those objects. These desired states can be received from other VOs, or the cloud layer. There may also be many virtual objects associated with one physical object. Virtual objects can hold the entire data set generated by their physical object, or subsets of that data. The virtual object layer in E-ACO architecture can have one or many cyber entities (virtual object or digital twins) for both clustered and individual objects. These twins can be created in the cloud layer, or local edge layer to support real time communication. For example, when sensors s$_{1}$ and s$_{2}$  across different clustered objects\ncommunicate with each other, the sequence of communication via\nvirtual object layer should follow starting s1 to vs$_{1}$  (digital twin of\ns$_{1}$), vs$_{1}$  to vs$_{2}$  and vs$_{2}$  to physical sensor s$_{2}$ . \n\n The \\textit{cloud layer} is the location of long term storage of device state, as well as more complex processing of received device data. Computationally intensive operations can be performed at this layer, thereby easing the burden of devices themselves as well as the hardware at the edge. These operations could include, but are not limited to: image processing with the intent of facial or object recognition, machine learning in order to fine tune a system's efficiency, or data visualizations. This layer manages communication with the virtual object and application layers, and is responsible for propagating control signals entered by the user as well as generating control signals based on the aforementioned data processing. Communication between clouds can also take place within this layer to enable big-data analytics or the union of discrete but related implementations. Single or multiple cloud scenarios can exist to support\nfederation or trusted collaboration between them. Some important IoT cloud platforms include Amazon AWS\\footnote{https:\/\/aws.amazon.com\/}, Microsoft\nAzure\\footnote{https:\/\/azure.microsoft.com\/en-us\/} IoT Hub, and Google Cloud IoT Core\\footnote{https:\/\/cloud.google.com\/iot-core}. An important use for cloud layer in IoT\/CPS involves defining security policies for authorized\ncommunication among different objects. \n\nThe \\textit{application layer}, is responsible for both displaying system information to the user and for user input. This layer needs to communicate with the layer directly below it to pass on control signals and receive visualizations and system state information. Users and administrators can remotely send commands and instructions to smart devices residing within the bottom layer using these applications, but such\ninteraction must propagate through the other two ACO middleware layers (cloud services and virtual object)\n\nThe layered access control oriented (ACO) structure discussed was proposed by Alsehri and Sandhu\\cite{7809752} with a focus on clarifying the middleware layers' function and form in IoT architectures. The distinction between the virtual object layer and the cloud layer lends itself to integration of heterogeneous objects into the system, as well as giving a well defined framework for access control techniques. This work also supports computation at the edge, as opposed to the cloud layer, by delineating the differences therein. Edge computation is necessary in industry with a focus on low latency that necessitate fast response times i.e. autonomous cars, or dynamic agricultural monitoring systems such as drones.\n\n\\subsection{Related Work}\n\nRecent extensive analysis of IoT technologies into the field of agriculture has been published by Gupta et al \\cite{gupta2020security}. The authors found that computation at the edge is a requirement for many systems with a focus on real time analysis and dynamic behavior. However, the assignment of responsibility at the edge comes with an increased attack surface due to the array of heterogeneous physical devices deployed\\cite{sina2020farming}. These devices are usually not designed with security as a chief concern\\cite{o2016insecurity}, and are a major security liability if configured incorrectly. The deployment of cryptographic security measures are difficult at the device level due to the computational constraints of most IoT enabled devices. While solutions do exist \\cite{dhanda_singh_jindal_2020}, they are relatively novel and have not yet found widespread implementation. They propose a lightweight multi-factor authentication protocol in the form of an independent Certificate Authority (CA). This allows for dynamic authentication and meets the complexity needs at the device level. It is worth noting however, that this solution does not detail practices to limit what data is being shared, only how to grant authorization. \n\nAnother area smart connectivity can greatly improve performance and efficiency is manufacturing. Kusiak\\cite{kusiak2018smart} makes the case that due to the trend of ever-increasing integration of smart sensors into manufacturing environments, the utilization of that data will drive further integration of smart actuators and data analysis into manufacturing processes. The employment of this novel data will lead to more accurate and complex modelling, optimization, and simulation. These models will give insight into potential fine-tuning practices to increase manufacturing efficiency, and the analysis of equipment monitoring will lead to predictive maintenance and prevention of equipment failure\\cite{s20195480}. This comes at the cost of increased cyber-security and safety concerns. As companies become reliant on modelling and IoT device infrastructure the value of these technologies goes up, therefore their security becomes paramount to continued profit and growth. In regard to safety, as automation and autonomous smart decision-making becomes integral to manufacturing centers, the responsibility of equipment to function correctly continuously shifts to lie upon the cyber-physical implementation.   \n\n\\section{Need for Edge Centric Secure Data Sharing}\n\\label{sec:need-for-edge}\n\nImplementations of IoT technologies at scale involve the generation of large quantities of data, which are used to affect system state by adjusting IoT actuators present in the system. The metrics for this state change are system specific but all systems require the sharing of data generated by local physical devices. This sharing can take place directly from virtual device to virtual device, virtual device to the local edge, or local edge to cloud. Which type of sharing takes place is determined by the level of computation necessary before the system state is affected. \n\nIn all aspects of device data sharing in a smart IoT connected system the local edge is critical and extensively utilized. These edge systems ensure\nlow latency and real time communication much needed in\nmost smart applications without bandwidth issues. In such scenarios, the edge plays a role in virtual device to virtual device sharing because all shadow clients in these systems reside on these local edge. Therefore even if the hardware of the edge is unneeded for computations more complex than device hardware can handle, the mechanisms of data sharing between virtual device clients still reside on, and are controlled by the edge which works as a middle man and relay the data. Virtual device to local edge sharing is required to facilitate computations exceeding physical device hardware, aggregation of device data in order to manage the system as a whole, or simply for comprehensive logging of system state. In the case of local edge to cloud data sharing, the local edge acts as a data pass-through in order to supply system information to cloud resources for computations that exceed local edge hardware capacity. These computations may include, but are not limited to, facial or object recognition, complex image processing, or machine learning algorithms. \n\nDue to the local edge's involvement in all data sharing which take place within an IoT system, the security of edge and the data it holds is of the utmost importance. The architecture proposed in this paper focuses on securing data in the system by managing the allocation of individual pieces of data into dynamic subsets based on tags. This is a form of TBAC with a focus on reducing the `distance' between tags assigned to data and the data itself. The implementations of TBAC currently present at the industry level utilize rules to tag data and independently apply tags to resources. This creates separation between the data and the tags applied to that data, as well as the containers that data will be placed within. We aim to improve this by directly applying tags to data and distributing data into digital twins based on those tags. Therefore each digital twin will have a set of tags defining what subset of data it will hold, and data will be distributed into each twin based on tags attached directly to that data. \n \n \\begin{figure}[t!]\n  \\centering\n  \\includegraphics[width=8cm, height=2.8in]{Diagrams\/External_Car_Communication.png}\n  \\caption{External Smart Car Communication}\n  \\label{Car_comm}\n\\end{figure}\n\\subsection{Motivating Use cases}\n\\subsubsection{Smart Cars and Intelligent Transportation}\n Smart vehicles require low-latency with high-volume data sharing. The internal network-connected sensors and actuators present in the car must be continuously sharing their data with the edge. This data is processed to allow functionality such as lane assistance systems, emergency collision avoidance, or full autonomous navigation. Externally, the car may be communicating with roadway infrastructure such as traffic lights, speed limit transmitters, or construction zone signalling shown in Figure \\ref{Car_comm}. Sharing data with other smart vehicles offers many benefits as well, in the form of automated lane merging protocols, increased speed limits due to increased reliability of surrounding vehicles, and shared awareness of roadway hazards. These factors culminate in smart cars prioritizing internal sensor-to-edge and external edge-to-edge sharing.\n\n While local and edge-to-edge sharing is prioritized, there is also utilization of the cloud layer in both logging data and implementation of more complex algorithms. User usage data such as location, driving habits, and maintenance history can be stored in the cloud for later retrieval. Performance data generated by the vehicle can also be sent to the cloud for processing by machine learning algorithms in order to monitor system health and send preemptive maintenance alerts. \n\n\\subsubsection{Smart Manufacturing}\n Smart manufacturing environments can take advantage of IoT technologies by distributing large quantities of internet connected smart sensors throughout the production pipeline. The local edge can be used to monitor system health by ensuring that sensor values fall within acceptable operating ranges. The cloud layer ensures system health by employing machine learning algorithms which monitor system efficiencies as reported by sensors in the system and give predictive points of failure. This architecture considers the necessity for low latency response times in the event of critical failure via utilization of the edge as a monitoring system, while also encouraging long-term health of the system via utilization of machine learning resources in the cloud. \n\n\\subsection{Threat Model}\n\\par The adversary threat model considered in this paper is heavily influenced by the security research put forward by the USDOT Intelligent Transportation Systems Office \\footnote{https:\/\/www.its.dot.gov\/factsheets\/cybersecurity.htm}. We have chosen to consider this research in developing our threat model because the environment it studies, smart transportation, is one of the most dynamic and difficult to secure. It is also the most industry applicable environments for IoT requiring edge based solution, as described earlier. The threats and vulnerabilities we address in the proposed solution include:\n\\noindent\n\\begin{itemize}[leftmargin=*]\n    \\item Entities authorized to read or affect system state of objects may get access to extraneous data which they should not have. As an example, roadway infrastructure such as speed limit transmitters should be allowed to affect maximum speed of a smart vehicle, but should not be able to read or write data such as location, personal user data, vehicle specifications, or maintenance information.  In traditional IoT digital twin architectures access is granted as a binary, where users are authorized to view and affect contents of a digital twin as a whole or not at all. This exposes even authorized entities to an excess of data, and is less secure than giving access to individually tagged pieces of data. \n    \\item Due to the large number of IoT devices in ecosystem such as smart factories, ITS, or smart homes, it is a near certainty that some of these devices will malfunction. In all of these objects failure may have severe consequences, therefore quick and efficient realization of device malfunction is a necessity. The attachment of tags directly to pieces of data allows for consistent processing and verification regarding the value of that data by the associated digital twin. For example, all values tagged 'temperature' within a system could have bounds implemented as rules such as: temperature should be a positive integer, and temperature should never exceed 100 units. If a piece of data exceeds or falls below these bounds then it is safe to assume that the physical device is malfunctioning and system state is compromised.\n\\end{itemize}\nThis paper proposes an edge based solution addressing these security concerns via data distribution into multiple digital twins foundationally built on TBAC. We also support and build upon security properties addressed by USDOT ITS research. We focus on \\textbf{Authenticity \\& Trust} by implementing open source software such as Mosquitto\\footnote{https:\/\/mosquitto.org\/} which maintains support for multiple forms of authorization including username\/password, PSK (Pre-Shared Key), and external plugin support. This allows for system specific authorization schemes to be implemented, while also providing built in authorization methods. \\textbf{Confidentiality \\& Privacy} is supported in this architecture by the subdivision of data into multiple digital twins. Data exposure is limited by allowing authorized entities to view only the subset of data they require to function, thereby keeping the information in the system confidential and private.\n\n\\subsection{Some Industry Solutions and Limitations}\n\\subsubsection{Microsoft Azure}\n Microsoft Azure IoT Hub allows attachment of tags to digital twins and physical devices but they are static informational metadata such as device specific location\/properties and do not serve a security function nor do they delineate pieces of data. Queries can be used to route data into digital twins based on tags, but are not dynamic and queries must be added to process additional tags. Digital twins in this architecture may not receive subsets of generated data as tags are applied to physical devices, not individual pieces of data. Therefore digital twins may be tagged in order to authorize reception of device data, however this authorization is purely a binary: either they will receive the full device message if tags are matching, or they will receive nothing. This limitation is not present in our implementation because the tags are attached to each key-value pair in every message and therefore messages may be subdivided based on tags.\n\\subsubsection{Google IoT Core}\nGoogle IoT Core offers a highly scalable industry IoT solution, however does not implement distribution of data based on tags. Tags in their architecture can be applied to physical devices and serve as device identifiers specifying metadata information such as: serial number, location, or manufacturer information. Tags may also be applied at the digital twin level in order to grant access to users authorized to view individual tags. However due to the lack of data distribution based on data present, all data is collected in one digital twin. Therefore subsets of data cannot be accessed and in order to view device data a user must be authorized to view all tags present. This is subversive to the limitation of data sharing in the system and is less dynamic than access control granted to individual shadows and therefore tags. \n\\subsubsection{AWS IoT Core}\nAmazon Web Services IoT Core supports a many-to-many digital twin-to-physical device relationship in the form of named digital twins accompanying a base unnamed twin. Physical devices may publish data directly to their named shadow counterparts, or publish all data to the base unnamed shadow which can then manage publications to named shadows. The purpose of named shadows is to hold subsets of physical device data in order to minimize data exposure and system malleability upon authorization of a resource regarding access to the shadow. This division of data comes closest to our proposed architecture, however there is no support for tagging discrete pieces of data. Rules can be implemented to distribute data to named shadows, however due to the lack of tag attachment to data these rules must work on data value, associated key, or other system information. This means data can be subdivided in the system, but like data can not be effectively grouped dynamically. Rules must be defined to sort individual data keys into named shadows resulting in a less scalable and more implementation specific system. \n\\subsubsection{Oracle IoT Asset Monitoring Cloud Service}\n Oracle's cloud IoT service allows the creation of digital twins to hold device information, as well as predictive twins to hold the results of complex analysis of device performance such as machine learning and neural networks. They also allow simple creation of rules regarding alerts and system functioning such as location-based rules which activate when a device enters or exits defined locations, threshold-based rules which trigger when a devices reported data either exceeds or falls below set values, and alert-based reactions which trigger physical device actions given alerts present in the system. However the tags which can be attached to devices are purely descriptive and serve no security or access control centric function. Therefore the division of data in this architecture is difficult, as individual pieces of data are not delineated in any way other than their associated keys. Highly dynamic environments may suffer security consequences as authentication in this architecture is a binary of full access or no access. \n\n\\section{Proposed Multiple Digital Twins with \\\\ Tags Based Access Control}\n\\label{sec:tbac-proposal}\n It is clear at this point that IoT environments generate and subsequently share large amounts of data. Mechanisms for sharing relevant and required information facilitate correct data apportionment between resources, as well as limiting the amount of data shared as much as possible. Minimizing data sharing within the architecture both increases security and decreases the burden on networking hardware. \n Our approach to controlling data sharing implements subdivision of data generated by physical sensors, and grants individual access to those subsets. This employs the security principle of least privilege by giving access to only the information required by the authorized resource, and allowing system malleability on the smallest surface possible. This increases system security as well as efficiency by minimizing the size of data flowing in the system from producers to consumers. \n\n\\begin{figure*}[t!]\n\\centering\n    \\includegraphics[scale=0.51]{Diagrams\/Sub-JSON-Generation-new-edited.png}\n    \\caption{Propagation of Reported States to Sub-JSONs}\n    \\label{sub_json_generation}\n\\end{figure*}\nDigital twins are the source of this subdivision, as they can exist in a many-to-one relationship with their physical counterparts. Each shadow instance holds a subset of the data present and can independently grant access to resources. These resources may query the shadow for the current system state, or publish desired states to the system. The resulting architecture leads to a distribution of data, and prevents a single MQTT client assuming all interaction with resources wishing to read or affect system state. The modularity of the separation of data into many separate digital twins also affords flexibility because not all clients must be active at any given point in time. Twins have the potential to be spun up or spun down as necessitated by resources in a form of load balancing. If a digital twin registers long periods of disconnection or inactivity from its associated device, the client could be halted until the device either has a state to report or the subset of information the client holds is requested by an external resource. This reactivity could be converted to a highly dynamic and scalable system which manages the number of active twins in real time based on demand. \n\nThe implementation of physical devices is as straightforward in this architecture as it is in a one-to-one device-to-twin structure. Due to the centralized nature of MQTT, physical devices need only subscribe to topics following a pre-defined API (Application Programming Interface) structure to receive state change control signals. Authorization to publish to those topics may be handled by the broker, giving a central point at which access control can be done regarding all digital twins. This ensures security of the channels in which interaction takes while requiring few subscriptions from the physical device.\n\n\\subsection{Proposed Architecture}\nThe distribution and subdivision of data in our architecture is facilitated by the application of tags. Each key-value pair in the system holds a key string describing the meaning of the data held in the object and a value array containing the sensor value and tags attached to the object. These tags identify the function of that data within the implementation, provide structure for groupings of related data, and are the central mechanism for access control.\nIn this architecture tags support grouping of data by allowing similar data to be quickly associated and divided into subsets. Figure \\ref{sub_json_generation} shows the processing of reported states (from the physical device to base shadow) with attached tags, and the division of data (from base shadow to multiple sub JSONs) based on those tags. For example in smart cars there are many different sets of data that could be produced such as speed, location, pressure, temperature, etc. All sensors in the car would then attach `pressure' to data measuring a pressure in the car. Additionally, more specific subsets can be made in order to grant external resources access to only the information they require. Therefore pressure data being monitored associated with the tires of the car may be tagged `tire' as well as `pressure' in order to differentiate it and allow more specific data sharing. Tags serve to group the data into most specific subset possible, after which the key associated denotes exactly what that data represents in the system.\n \nTags and key-value pairs hold a many-to-many relationship where one tag may be applied to many key-value pairs and conversely one key-value pair may hold many tags. This relationship allows data values to be distributed and held by many shadows, and also one shadow may receive many data values at once if a single tag is distributed to multiple key-value pairs in a message. As discussed earlier, where many key-value pairs are tagged `pressure' in a single message and subsequently distributed to the 'pressure' digital twin.\n\\begin{figure}[t!]\n \n  \\includegraphics[width=\\linewidth]{Diagrams\/System_Architecture-edited.png}\n  \\caption{Implemented System Architecture}\n  \\label{des_state_resolution}\n\\end{figure}\n\\subsection{Assignment of Tags}\n The assignment of tags within a TBAC architecture must follow proper security practices, as assigned tags are the basis of access control. If tags are improperly assigned and therefore data is distributed to digital twins in which it should not reside, then resources that are given access to those twins will be served data they are not authorized to view. The attachment of low-security classification tags to high-security pieces of data is a simple way to gain access to critical data within the system. For example, if there exists a `timing' tag that functions as a benchmark to synchronize elements of the system then all resources would be able to access the digital twin containing `timing' information. If administrator is able to attach the `timing' tag to a piece of sensitive information such as location, or user data, it will enable unauthorized data read via the `timing' digital twin. \n\n Tags should only be malleable to a few key authenticated resources in the system. They may be applied by the physical device itself based on characteristics of the data being generated. This is the foundation of on-the-fly dynamic tag attachment within the architecture. For example, a smart temperature sensor may have a ceiling at which the recorded temperature is no longer safe. When the recorded temperature exceeds that ceiling a `warning' tag could be applied to the data in order to trigger a safety system or inform the user. Additional levels may be present as well, so if the temperature exceeds another threshold a `critical' tag may be applied. Therefore response behavior can vary dependent upon the level of device failure. These tags can be attached when a value exceeds or falls below a predefined set-point and are device specific. The attachment of these tags means the associated key-value pair will be placed within the `warning' or `critical' digital twins, which allows all system health to be monitored via a small number of digital twins. This functionality is shown in Figure \\ref{sub_json_generation}, where both tire pressure sensors are reporting values which have system health tags attached. The driver-side sensor has applied a `warning' tag which may be applied when the tire falls below the recommended specification by a relatively small margin. The passenger-side sensor has applied a `critical' tag which may be applied when the pressure falls too far e.g. below 30. \n\n System administrators should also be able to add tags to device data as necessary. Therefore tags published to the device by the base unnamed shadow are applied to all further data generated by the device. Only the base shadow should have this functionality as it is the most controlled due to the centralized nature of the data it holds, and therefore administrators should be the only users with access. \n\n\n\\begin{table}[t!]\n\\begin{center}\n\\caption{Raspberry Pi 4 Model B Specifications}\n\\renewcommand{\\arraystretch}{1.5}\n \\begin{tabular}{ | p{0.60in} | p{2.25in} | } \n \\hline\n Operating System & Raspberry Pi OS, May 7, 2021 \\\\ \n \\hline\n CPU & Broadcom BCM2711, Quad core Cortex-A72 (ARM v8) 64-bit SoC @ 1.5GHz \\\\\n \\hline\n RAM & 4 GB LPDDR4-3200 SDRAM \\\\\n \\hline \n Network Interface & Gigabit Ethernet, 2.4GHz and 5GHz 802.11b\/g\/n\/ac Wi-Fi \\\\\n \\hline\n\\end{tabular}\n\\label{tab:Raspberry-pi-specs}\n\\end{center}\n\\end{table}\n\n\\section{Implementation and Evaluation}\n\\label{sec:evaluation}\n\n\\begin{figure}[t!]\n  \\centering\n  \\includegraphics[width=7cm, height=3.4in]{Diagrams\/Experimentation_Struture-edited-v2.png}\n  \\caption{Transformation from Five to Six Key-Value Pairs}\n  \\label{experiment_state_transform}\n\\end{figure}\nThe proposed architecture has been evaluated using a lightweight digital twin implementation written in python and emulating AWS MQTT topic structure. This allows for large scalability and integration into industrial environments with minimal modification of the current code base. In this section, mechanism of interaction with devices in this architecture will be explained, then the performance of our implementation will be evaluated, finally application to industry will be discussed.\n\n Device state is divided into three subgroups: \\textit{reported}, \\textit{desired}, and \\textit{delta}. The physical device modifies the reported subgroup when it is connected and reports its current state. Desired states may be pushed to the digital twin created in the local edge by authorized external clients in order to request a change in physical device state. If a difference is determined between the reported and desired states of the device, then the differing keys are added to the delta subgroup. When the device is connected, the calculated delta state is published to the physical device. Once the device receives these keys and transitions state, it reports the new state. Upon reception of a reported state matching a given desired state, the digital twin acknowledges that state as resolved and removes the associated key from both the desired and delta subgroups. The key-value pairs present in the reported state are then divided based on attached tags and distributed to their associated base shadows in the local edge, as shown in Fig. \\ref{des_state_resolution}. The shadow and device python clients were run on a Raspberry Pi 4 for data collection, the specifications of which can be seen in Table \\ref{tab:Raspberry-pi-specs}. The open source broker Mosquitto\\footnote{https:\/\/mosquitto.org\/} was used for authentication as it includes username\/password, PSK (Pre-Shared Key), and external plugin support. The system was not stressed with the entirety of the architecture running on one device, and this should not affect the timing data collected as only the processing time of the digital twin is being evaluated.\n\n\n\n\n\n\n\\begin{figure}[t!]\n  \\centering\n  \\includegraphics[width=\\linewidth, height=2.15in]{Diagrams\/Dynamic_graph_pi_99ci-edited.png}\n  \\caption{Average Dynamic Tag Assignment Processing Time}\n  \\label{dynamic_processing_time_graph}\n\\end{figure}\n\\begin{figure}[t!]\n  \\centering\n  \\includegraphics[width=\\linewidth, height=2.5in]{Diagrams\/static_graph_pi-edited.png}\n  \\caption{Average Static Tag Assignment Processing Time}\n  \\label{static_processing_time_graph}\n\\end{figure}\n\nIn order to evaluate the system's performance, the shadow linearly scales the relationship between the number of key-value pairs and tags associated with each pair in the system at a given point, as shown in Fig. \\ref{experiment_state_transform}. For example, when there are five key-value pairs in the system there will be five tags attached to each of those pairs. Upon reception of a reported state from the simulated device, the shadow introduces a new key-value pair as well as increments the number of tags attached to all other pairs previously present. Therefore once the five pair state is reported by the device, a sixth pair is added and a new tag is appended to each of the five pairs already present. The new desired state of the system is then added to delta subgroup and subsequently published to simulated device. Once the simulated device conforms to the received desired state and reports its current state, this cycle continues. \nTo effectively measure the efficiency of our architecture, we calculate time only while the shadow is resolving messages. This avoids introduction of variance from network delay as well as abstracts our evaluation away from device specific implementations. Different IoT devices will have widely varying computational hardware as well as polling rates depending on implementation specific variables. The exclusion of device interaction from collected timings leads to more consistent results and a stronger examination of our implementation of the proposed architecture.\n\n\n\n\nThere are three distinct function calls included in the timings collected: `update', `delta', and `parse\\_tags'. Update function processes the incoming message and places relevant key-value pairs into their designated position within the JSON structure. The delta function balances the JSON structure to ensure it retains continuity and consistency regarding the AWS-style. This encompasses functionality discussed earlier, such as removing keys from desired subgroup once a matching reported state has been received. Once the JSON has been cleaned the remaining keys in delta subgroup are published to device. The final call is to `parse\\_tags', which compiles a list of all tags attached to key-value pairs in the message and generates and publishes sub-JSONs to associated shadows.\n\n \n\n\n\nFigure \\ref{dynamic_processing_time_graph} shows the processing time of the system averaged over 500 trials where each trial incremented the number of key-value pairs in the system from zero to forty and then emptied the system. Each point is the average performance time of the system associated with that many key-value pairs present. The attached error bars indicate a 99\\% confidence interval and show that with 1600 total tags present in the system (40 attached to 40 key-value pairs), the average processing time will rarely exceed 36 ms.\n Figure \\ref{static_processing_time_graph} displays the variation in system function with a static number of tags attached to each key-value pair present. Each data series represents an incrementation of number of key-value pairs from zero to one hundred with each additional pair containing specified number of tags, e.g. at data point 80 in the `5 Tags' series there are 80 key-value pairs each of which has 5 attached tags. Each data series is the average of 500 trials where each trial represents the filling of the system from zero to one hundred pairs. \n\n\n\nThese results are promising for real time and edge centric industry applications as they show minimal increase in processing time at the digital twin level, while largely scaling the number of tags and data pairs present in the system. Due to the exponential nature of the `parse\\_tags' function, it is the largest bottleneck of the system. However, if the number of tags applied to each key-value pair is kept low the system remains scalable and suffers minor performative degradation. \n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nThis research demonstrates edge centric access control structure in IoT environments by proposing a novel TBAC architecture focused on the division of data into multiple digital twins. This architecture fills a gap in environments where on the fly and real time limited data exposure is highly critical, and allows for complete subdivision of data based on tags attached directly to present data. Complex device relationships are supported via the many-to-many relationship between tags and data, allowing implementations to model peculiar environments with little additional complexity. We discuss the usefulness of this architecture in smart environments such as manufacturing and internet-connected vehicles, and give an example of the flow of tagged data in these environment. Industry solutions currently offered have been examined regarding their integration of TBAC as well as their capacity to divide data into subsets. The weaknesses and strengths of offered services are discussed in relation to the proposed architecture. We deployed a local implementation of our architecture and examined the effect of number of attached tags on performance. We envision further exploration regarding access control on the tagged shadows, and the application of this data distribution to other smart environments. \n\n\n\\section*{Acknowledgement}\nThis research is supported by NSF CREST Center Grant\nHRD-1736209 at UTSA, and by the Grant 2025682 at TTU.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nNon--linear integrable differential systems have been subject to intensive\ninvestigations since last century. In the last few years,\none of the most exciting\ndevelopments in this field is the revelation of its close relationship\nwith 2--dimensional exactly sovable field theories, in particular\n2--dimensional quantum gravity  and string theory. This relation enables us to extract\nnon--perturbative properties of non--critical string. More precisely,\nthe discretization of 2--dimensional quantum gravity  can be reformulated as matrix models. Through\nthe {\\it double scaling limit}, one can prove that 1--matrix model  with even\npotential is described by {KdV hierarchy } and string equation.\nRecently it has been shown that the double scaling limit is not an inevitable\nstep. In other words, the {KdV hierarchy } is not merely an occasional effect of double\nscaling limit but intrinsic in matrix model formulation.\nThe idea is as follows: we represent matrix models as certain\ndiscrete linear\nsystem(s), from which we can extract lattice integrable hierarchies, finally\nwe can directly extract differential hierarchies from these lattice\nhierarchies. Using this approach, one can prove that\nthe 1--matrix model  with general potential is characterized by { non--linear Schr\\\"odinger hierarchy }(NLS), which is\nthe two bosonic field representation of { KP hierarchy }.  In other words, the 1--matrix model  gives\na new solution for  the $\\tau$--function  of { KP hierarchy }\\cite{BX1}.\nIn the same way, we can apply this procedure to multi--matrix models  to obtain their\nfull differential integrable hierarchies. In this process, we are led to\nconsider a possible generalization of { KP hierarchy }. The study of this generalization\nis the main purpose of this letter. Actually another enlargement of { KP hierarchy } was\nconsidered some years ago by adding additional flows. However, although\neach of the additional flows commutes with all the old KP flows, they don't\ncommute among themselves\\cite{Dickey}. Thus this generalization is not\nconsistent. Fortunately, we will see that properly introducing new flows\nwe can obtain an integrable hierarchy, in which all the flows are\ncommutative. We will explain the realization of such generalized { KP hierarchy }\nin multi--matrix models  elsewhere\\cite{BX2}.\n\n\n\n\\section{KP hierarchy}\n\nLet us begin with a pseudo--differential operator(PDO) of arbitrary order\n\\begin{eqnarray}\nA=\\sum_{-\\infty}^n a_i(x)\\partial_i.\\label{pseudodp}\n\\end{eqnarray}\nwhere $x$ is {\\it the space coordinate}, while\n${\\partial}^{-1}$ is formal integral operation over $x$.\nAll the pseudo--differential operators form an algebra $\\wp$ under the\ngeneralized Leibnitz rules\n\\begin{eqnarray}\n&&{\\partial} a(x)=a(x){\\partial}+a'(x),\\qquad [{\\partial}, x]=1,\\nonumber\\\\\n&&\\partial^{-1}\\partial=\\partial\\partial^{-1}=1,\\label{leibnitz}\\\\\n&&\\partial^{-j-1}a(x)=\\sum_{l=0}^{\\infty}(-1)^l{{j+l}\\choose l}\na^{(l)}(x)\\partial^{-j-l-1},\\nonumber\n\\end{eqnarray}\nwhere $a^{(l)}(x)$ denotes ${{{\\partial}^l a(x)}\\over{{\\partial} x^l}}$.\nThe algebra $\\wp$ has two sub--algebras:\n\\begin{eqnarray}\n\\wp=\\wp_+\\oplus\\wp_-.\\nonumber\n\\end{eqnarray}\nwhere $\\wp_+$ denotes the algebra of pure differential operators,\nwhile $\\wp_-$ means the algebra of pure integration operations.\n\nFor any given pseudo--differential operator $A$ of type (\\ref{pseudodp}), we\ncall $a_{-1}(x)$ its residue, denoted by\n\\begin{eqnarray}\n\\rm res_{{\\partial}}A=a_{-1}(x)\\qquad \\hbox{ or}\\qquad A_{(-1)}\\nonumber\n\\end{eqnarray}\nand we define the following functional\n\\begin{eqnarray}\n<A>=\\int a_{-1}(x)dx.\\label{innerproduct}\n\\end{eqnarray}\nwhich naturally gives an inner scalar product on the algebra\n$\\wp$\\cite{Babelon}.\n\n\\subsection{The integrable structure}\n\nNow let $L$ be a pseudo--differential operator of the first order\n\\begin{eqnarray}\nL={\\partial}+\\sum_{i=0}^{\\infty}u_i(x){\\partial}^{-i}\\label{PDO}\n\\end{eqnarray}\nwhich we will call KP(or Lax) {\\it operator}.\nWe call $u_i$'s KP {\\it coordinates}.\n$(L-{\\partial})\\in \\wp_-$, so we can represent\n a functional of KP coordinates as\n\\begin{eqnarray}\nf_X(L)=<LX>,\\qquad\\qquad X\\in\\wp_+\\nonumber\n\\end{eqnarray}\n\\def${\\cal F}(\\wp_-)$ {${\\cal F}(\\wp_-)$ }\nwhich  span a functional space ${\\cal F}(\\wp_-)$ . The remarkable fact is that ${\\cal F}(\\wp_-)$  is\ninvariant under the co--adjoint action of $\\wp_+$, consequently the algebraic\nstructure on $\\wp_+$ determines the Poisson structure on ${\\cal F}(\\wp_-)$ \n\\begin{eqnarray}\n\\{f_X, f_Y\\}_1(L)=L([X,Y])\\label{poisson1}\n\\end{eqnarray}\nThe infinite many conserved quantities (or Hamiltonians) are\n\\begin{eqnarray}\nH_r={1\\over r}<L^r>\\qquad \\forall r\\geq1\\label{hamiltonian}\n\\end{eqnarray}\nThey generate infinite many flows,\n\\begin{eqnarray}\n\\ddt r L=[L^r_+, L]\\label{KPequation}\n\\end{eqnarray}\nwhere the subindex ``+\" indicates choosing the non--negative powers of ${\\partial}$.\nSince\n\\begin{eqnarray}\n[L^r_+, L]=[L, L^r_-]\\in{\\cal P}_-,\\nonumber\n\\end{eqnarray}\nwe see that all the flows preserve the form of KP operator ``$L$\", and\nthey all commute with each other. This commutativity implies\nthe ``{\\it zero curvature representation}\"\n\\begin{eqnarray}\n\\ddt m L^n_+-\\ddt n L^m_+=[L^m_+, L^n_+],\\qquad \\forall n,m\n\\label{zerocurvature}\n\\end{eqnarray}\nBy { KP hierarchy } we mean the set of differential equations (\\ref{KPequation}) or\n(\\ref{zerocurvature}).\nIn fact the { KP hierarchy } possesses another Poisson structure\\cite{Watanabe}\n\\begin{eqnarray}\n\\{f_X, f_Y\\}_2(L)&=&<(XL)_+YL>-<(LX)_+LY>\\nonumber\\\\\n&+&\\int [L,Y]_{(-1)}\\Bigl({\\partial}^{-1}[L,X]_{(-1)}\\Bigl)\n\\label{poisson2}\n\\end{eqnarray}\nWith respect to these Poisson brackets,\nthe KP coordinates $u_i$ form $W$--infinity algebras.\nThe important point is that these two Poisson structures are compatible\nin the sense\n\\begin{eqnarray}\n\\{f, H_{r+1}\\}_1=\\{f, H_r\\}_2\\qquad \\forall\\qquad\\hbox{function f}\n\\end{eqnarray}\nThis compatibility ensures the integrability of { KP hierarchy }. Generally speaking,\nfor a system of infinite many degrees of freedom, for example, the { KP hierarchy },\nwe may  find various definitions of integrability\\cite{Babelon}.\nThe essential point is that there must exist infinite many conserved\nquantities in involution. Therefore we can list  some of the definitions\nbelow\n\\begin{enumerate}\n\\item\n{\\it There exist two compatible Poisson brackets}\n({\\it or bi-hamiltonian structure}).\n\\item\n{\\it The flows are all commutative}.\n\\item\n{\\it There  exists the zero curvature representation}(\\ref{zerocurvature}).\n\\end{enumerate}\nFor different purposes we may use different definitions. For example,\nbi-Hamiltonian structure can exhibit the  Poisson algebraic structure of the\nsystem. But in the next two sections we will mainly use the second definition\nto prove the integrability of the generalized { KP hierarchy } due to its simplicity.\n\n\n\\subsection{The associated linear system}\n\nThe KP operator $L$ can be expressed in terms of the ``dressing\" operator\n\\begin{eqnarray}\nL=K{\\partial} K^{-1}\\qquad K=1+\\sum_{i=1}^{\\infty}w_i{\\partial}^{-i}\\nonumber\n\\end{eqnarray}\nAfter defining\n\\begin{eqnarray}\n\\xi(t,\\lambda)=\\sum_{r=1}^{\\infty}t_r\\lambda^r\n\\end{eqnarray}\nand introducing the Baker--Akhiezer function\n\\begin{eqnarray}\n\\Psi(t,\\lambda)=Ke^{\\xi(t,\\lambda)}\\label{baker}\n\\end{eqnarray}\nwe can associate to the KP hierarchy (\\ref{KPequation}) a linear system\n\\begin{eqnarray}\n\\left\\{\\begin{array}{l}\nL\\Psi=\\lambda\\Psi,\\\\\\noalign{\\vskip10pt}\n\\ddt r \\Psi=L^r_+\\Psi.\n\\end{array}\\right.\n\\label{flow}\n\\end{eqnarray}\n\n\\subsection{ The $\\tau$--function}\n\nOne of the important ingrediants of the KP system is its $\\tau$--function,\nwhich can be introduced through Baker--Akhiezer function\n\\begin{eqnarray}\n\\Psi(t,\\lambda)={{\\tau(t_1-{1\\over {\\lambda}},t_2-{1\\over\n{2\\lambda^2}},\\ldots)}\n\\over{\\tau(t)}}e^{\\xi(t,\\lambda)}\n\\end{eqnarray}\nOne can prove that\n\\begin{eqnarray}\n{{\\partial}^2\\over{{\\partial} t_1{\\partial} t_r}}\\ln\\tau=\\rm res_{{\\partial}}L^r,\\qquad \\forall r\\geq1\n\\end{eqnarray}\nIf we define a set of new functionals\n\\begin{eqnarray}\nJ_r=\\int \\rm res_{{\\partial}}L^r dx,\\qquad \\forall r\\geq1\n\\end{eqnarray}\nthen\n\\begin{eqnarray}\n \\ddt s J_r=0,\\qquad\\quad \\forall r,s\\geq1\n\\end{eqnarray}\nSo $J_r$'s are the conservation laws of the KP hierarchy.\n\n\\section{Generalization of KP hierarchy }\n\n\\setcounter{equation}{0}\n\\setcounter{subsection}{0}\n\\setcounter{footnote}{0}\n\nNow we come to discuss the generalization of the { KP hierarchy }, which we promised\nin the introduction.\n\n\\subsection{The additional flows}\n\nOur purpose is to show\nthat we may introduce other series of flows. In order to do so, we define\na new operator\\cite{Chenlee},\n\\begin{eqnarray}\nM\\equiv K(\\sum_{i=1}^{\\infty}rt_r{\\partial}^{r-1})K^{-1}=\\sum_{i=-\\infty}^{\\infty}\nv_i{\\partial}^i.\\label{mpdo}\n\\end{eqnarray}\nwhich is conjugate to the KP operator $L$ in the sense that\n\\begin{eqnarray}\n[L, M]=K [{\\partial}, \\sum_{i=1}^{\\infty}rt_r{\\partial}^{r-1}]K^{-1}=1,\\label{conjugate}\n\\end{eqnarray}\nWe can derive the equations of motion for $M$,\n\\begin{eqnarray}\n\\left\\{\\begin{array}{l}\n\\ddt r M=[L^r_+, M],\\\\\\noalign{\\vskip10pt}\n{\\partial \\over {\\partial \\lm}}\\Psi=M\\Psi.\n\\end{array}\\right.\n\\end{eqnarray}\nSo we see that $L$ and $M$ are nothing but the operatorial expressions of\n${\\lambda},{\\partial \\over {\\partial \\lm}}$ (acting on  $\\Psi(t,{\\lambda})$).\n\nAs we know, the basic requirement for new flows is that they should preserve\nthe form of KP operator $L$. So we can define new flows like\n\\begin{eqnarray}\n\\ddt {mn} L=[L, (M^mL^n)_-],\\qquad\\forall m,n.\\label{additional}\n\\end{eqnarray}\nOne can show that each flow commutes with KP flows (\\ref{KPequation}), but\nthese additional flows do not commute among themselves\\cite{Dickey}. Our aim\nis to show that properly choosing combinations of these additional flows, we\ncan define new flows which commute with the old KP flows and among themselves.\n\n\\subsection{Another series of flows}\n\nOur  starting  remark is that the $t$--series of perturbations is due to\nthe fact that $[L, L^r_-]\\in{\\cal P}_-,\\forall r\\geq1$. Now\nwe also have $[L, M^r_-]\\in{\\cal P}_-,\\forall r\\geq1$,\nso we could introduce a new series of deformation\nparameters\\footnote{That is to say, the KP coordinates $u_i$'s and $v_i$'s\ndepend on both $t$ and $y$.} $y_1,y_2,y_3,\\ldots$, such that\n\\begin{eqnarray}\n\\left\\{\\begin{array}{l}\n\\ddy r L=[L, M^r_-],\\\\\\noalign{\\vskip10pt}\n\\ddy r \\Psi=-M^r_-\\Psi.\n\\end{array}\\right.\n\\end{eqnarray}\nAll of these equations together result in the following enlarged KP system\n\\begin{eqnarray}\n\\left\\{\\begin{array}{l}\n\\ddt r L=[L^r_+, L],\\\\\\noalign{\\vskip10pt}\n\\ddt r M=[L^r_+, M],\\\\\\noalign{\\vskip10pt}\n\\ddy r L=[L, M^r_-],\\\\\\noalign{\\vskip10pt}\n\\ddy r M=[M^r_+, M].\n\\end{array}\\right.\\label{gen}\n\\end{eqnarray}\n\nNow we  should prove  that these new series of perturbations do not distroy\nconsistency, that is to say,\nwe should check the commutativity of all these flows. In the following we only\nconsider an example, i.e.\n\\begin{eqnarray}\n\\ddt r\\bigl(\\ddy s L\\bigl)=\\ddy s\\bigl(\\ddt r L\\bigl).\\label{comflows}\n\\end{eqnarray}\nUsing eqs.(\\ref{gen}), we see that the left hand side is\n\\begin{eqnarray}\n&&\\quad \\ddt r\\bigl(\\ddy s L\\bigl)=\\ddt r[L, M^s_-]\\nonumber\\\\\n&&=[[L^r_+, L], M^s_-]+[L,[L^r_+, M^s]_-]\\nonumber\\\\\n&&=[[L^r_+, L], M^s_-]+[L,[L^r_+, M^s_-]]-[L,[L^r_+, M^s_-]_+]\\nonumber\\\\\n&&=[L^r_+, [L, M^s_-]]+[[L^r,M^s_-]_+, L]\\nonumber\\\\\n&&={\\rm r.h.s.}\\nonumber\n\\end{eqnarray}\nThe other cases can be checked in the similar way.\nSo the perturbations we introduced before indeed give an enlarged { KP hierarchy }.\nIts associated linear system is\n\\begin{eqnarray}\n\\left\\{\\begin{array}{l}\nL\\Psi={\\lambda}\\Psi,\\\\\\noalign{\\vskip10pt}\n\\ddt r \\Psi=L^r_+\\Psi,\\\\\\noalign{\\vskip10pt}\n\\ddy r \\Psi=-M^r_-\\Psi,\\\\\\noalign{\\vskip10pt}\nM\\Psi={\\partial \\over {\\partial \\lm}}\\Psi.\n\\end{array}\\right.\\label{lmpsi}\n\\end{eqnarray}\n\nThe usual KP hierarchy (\\ref{KPequation}) is a particular case\nof eqs.(\\ref{gen}) by fixing the $y$--series of the perturbations.\n\n\\subsection{The new basic derivative and the new bi--hamiltonian structure}\n\nAs we remarked a moment ago, when we disgard the $y$--series of flows, we\nrecover the usual { KP hierarchy }, whose hamiltonians are\n\\begin{eqnarray}\nH_{r(L)}={1\\over r}<L^r>,\\nonumber\n\\end{eqnarray}\nhere we use the subindex $(L)$ to indicate that the Hamiltonians are\nconstructed\nfrom the KP operator $L$. We may  also use the same symbol to denote the\nPoisson\nbrackets, $\\{, \\}_{(L)}$.\n\nNow if we fix all the $t$--series of parameters,  then we get another subset of\nthe enlarged hierarchy (\\ref{gen}), that is\n\\begin{eqnarray}\n\\left\\{\\begin{array}{l}\n\\ddy r L=[L, M^r_-],\\\\\\noalign{\\vskip10pt}\n\\ddy r M=[M^r_+, M].\n\\end{array}\\right.\\label{geny}\n\\end{eqnarray}\nThe second equation is in fact a { KP hierarchy } with KP operator $M$ of the form\n(\\ref{mpdo}). Since all these flows commute, this is an integrable\nsystem, and there should  exist two compatible Poisson brackets written\nin terms of coordinates $v_i$'s. However, this bi--hamiltonian structure\nis unknown due to the  fact that the positive powers of ${\\partial}$ in $M$ go to\ninfinity.\n\n\nFortunately, we may overcome the difficulty by introducing a new basic\nderivative. To this end, we recall that in our\nprevious analysis we treated $t_1$ as the space coordinate.\nFor later convenience, we denote $\\ddy 1$ by ${\\tilde\\d}$.\n{}From the $y_1$--flows of $\\Psi$, we may extract an operator identity\n\\begin{eqnarray}\n{\\tilde\\d}=-M_-=\\sum_{i=1}^{\\infty}\\Gamma_i{\\partial}^{-i}.\\label{dtilde}\n\\end{eqnarray}\nSince any positive powers of ${\\tilde\\d}$ belongs\nto ${\\cal P}_-({\\partial})$, so $\\{{\\tilde\\d}^i;i\\geq1\\}$\nforms a basis of ${\\cal P}_-({\\partial})$.\nWe may invert the relation (\\ref{dtilde}) to express ${\\partial}$ in\nterms of the new derivative\\footnote{\nRigorously speaking, this is only true when it acts on the function $\\Psi$.\nBut we  may think of it in the following way, starting from\n\\begin{eqnarray}\n{\\tilde\\d}\\Psi=\\sum_{i=1}^{\\infty}\\Gamma_i{\\partial}^{-i}\\Psi.\\nonumber\n\\end{eqnarray}\nproperly choosing the combinations of ${\\tilde\\d}$ such that we can\nreexpress the ${\\partial}^{-1}\\Psi$ in terms of new derivatives\n${\\tilde\\d}$, we replace  all the derivatives ${\\partial}$ in the linear system\n(\\ref{lmpsi}) by ${\\tilde\\d}$. So we may interpret $y_1$ as another space\ncoordinate.} ${\\tilde\\d}$\n\\begin{eqnarray}\n{\\partial}=\\sum_{i=1}^{\\infty}{\\tilde \\Gamma}_i{\\tilde\\d}^{-i}.\\label{d}\n\\end{eqnarray}\nUsing this fact, we get\n\\begin{eqnarray}\nM=-({\\tilde\\d}+\\sum_{i=1}^{\\infty}{\\tilde v}_i{\\tilde\\d}^{-i})=-{\\tilde K}{\\tilde\\d}{\\tilde K}^{-1}.\n\\end{eqnarray}\nwith new dressing operator ${\\tilde K}$ and new KP coordinates ${\\tilde v}_i$'s.\nObviously\n\\begin{eqnarray}\nM^r_-({\\partial})=M^r_+({\\tilde\\d}),\\qquad \\forall r\\geq1,\n\\end{eqnarray}\nwhere LHS is expanded in powers of ${\\partial}$, while the RHS is expanded in\npowers of ${\\tilde\\d}$.  Using eq.(\\ref{d}),\nwe can reexpress all the formulas (\\ref{gen}) in terms of this\nnew derivative ${\\tilde\\d}$, i.e.\n\\begin{eqnarray}\n\\left\\{\\begin{array}{l}\n\\ddy r (-M({\\tilde\\d}))=(-1)^{r+1}[(-M)^r_+({\\tilde\\d}),\n(-M)({\\tilde\\d})],\\\\\\noalign{\\vskip10pt}\n\\ddy r L({\\tilde\\d})=(-1)^{r+1}[(-M)^r_+({\\tilde\\d}), L({\\tilde\\d})],\\\\\\noalign{\\vskip10pt}\n\\ddt r (-M)({\\tilde\\d})=-[(-M)({\\tilde\\d}), L^r_-({\\tilde\\d})],\\\\\\noalign{\\vskip10pt}\n\\ddt r L({\\tilde\\d})=[L^r_+({\\tilde\\d}), L({\\tilde\\d})].\n\\end{array}\\right.\n\\end{eqnarray}\nApart from some additional signs, these equations are isomorphic to\neqs.(\\ref{gen}).\nThis reminds us that we can even consider $(-M)$ as a KP operator, and\nalternatively interpret $y_1$ as space coordinate, all the other parameters\nas time parameters.\nTherefore we can define two compatible Poisson brackets\nfor KP operator $(-M)$ by simply replacing $L$ in (\\ref{poisson1}) and\n(\\ref{poisson2}) by $(-M)$, which shows that\non the space $y_1$, the fields ${\\tilde v}_i$'s form\n$W_{\\infty}$ algebras too.\n\n\\section{Further perturbations and the full generalized { KP hierarchy }}\n\n\\setcounter{equation}{0}\n\\setcounter{subsection}{0}\n\\setcounter{footnote}{0}\n\nIn the previous section we have shown that the { KP hierarchy } can be perturbed by\nthe conjugate operator $M$ of the KP operator $L$.\nIn fact, the KP system allows further deformations.\n\n\\subsection{The new series of the flows}\n\nIn order to explain the further perturbations just mentioned,\nwe change a little bit our notation. Denote $t_r$'s and $y_r$'s  by\n$t_{1r}$ and $t_{2r}$ respectively. Furthermore define\n\\begin{eqnarray}\n&&L(1)\\equiv L,\\qquad\\qquad V(1)\\equiv\\sum_{r=1}^{\\infty}rt_{1r}L^{r-1}(1)\\nonumber\\\\\n&&L(2)\\equiv -{1\\over {c_{12}}}M\\qquad V(2)\\equiv\\sum_{r=1}^{\\infty}\nrt_{2r}L^{r-1}(2)\\nonumber\n\\end{eqnarray}\nNow let us introduce new operators in the following way\n\\ai\n&&L(\\alpha)\\equiv-{1\\over c_{\\alpha-1,\\alpha}}\\Bigl(c_{\\alpha-2,\\alpha-1}\nL(\\alpha-2)+V(\\alpha-1)\\Bigl)\\label{loperator}\\\\\n&&V(\\alpha)=\\sum_{r=1}^{\\infty}rt_{\\alpha,r}L^{r-1}(\\alpha),\\qquad \\alpha=3,4,\n\\ldots,n\n\\bj\nwhere $c_{{\\alpha},{\\alpha}+1}$'s are arbitrary constants, which amount to rescaling\n the space coordinates, and $n$ is an arbitrary positive integer.\nThen, in the same way, we can perturb the system further as follows\n\\ai\n&&\\ddt {\\beta r} L(\\alpha)=[L^r_+(\\beta), L(\\alpha)],\\qquad 1\\leq\\beta<\\alpha\n\\label{flowa}\\\\\n&&\\ddt {\\beta r} L(\\alpha)=[L(\\alpha), L^r_-(\\beta)],\\qquad\n\\alpha\\leq\\beta\\leq n\\label{flowb}\n\\bj\nNow in order to justify the consistency of these perturbations, we once again\nshould prove that all the flows commute among themselves. Let us check one\nexample,\n\\begin{eqnarray}\n\\ddt{{\\alpha} l}\\bigl(\\ddt{\\beta m}L(\\gamma)\\bigl)\n=\\ddt{\\beta m}\\bigl(\\ddt{{\\alpha} l}L(\\gamma)\\bigl),\\qquad{\\alpha}<\\beta<\\gamma.\\nonumber\n\\end{eqnarray}\nUsing the above hierarchy and Jacobi identities, we see that\n\\begin{eqnarray}\n{\\rm l.h.s.}=\\ddt{{\\alpha} l}[L^m_+(\\beta), L(\\gamma)]\n=[[L^l_+({\\alpha}), L^m(\\beta)], L(\\gamma)]+[L^m_+(\\beta), [L^l_+({\\alpha}),\nL(\\gamma)]].\\nonumber\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n{\\rm r.h.s.}=\\ddt{\\beta m}[L^l_+({\\alpha}), L(\\gamma)]\n=[[L^l({\\alpha}), L^m_-(\\beta)]_+, L(\\gamma)]+[L^l_+({\\alpha}),\n[L^m_+(\\beta), L(\\gamma)]].\\nonumber\n\\end{eqnarray}\nThe first term of ``l.h.s\" can be written as\n\\begin{eqnarray}\n{\\rm the~~1st~~term}&=&[[L^l_+({\\alpha}), L^m_+(\\beta)], L(\\gamma)]\n+[[L^l_+({\\alpha}), L^m_-(\\beta)]_+, L(\\gamma)]\\nonumber\\\\\n&=&[[L^l_+({\\alpha}), L^m_+(\\beta)], L(\\gamma)]+\n[[L^l({\\alpha}), L^m_-(\\beta)]_+, L(\\gamma)],\\nonumber\n\\end{eqnarray}\ntherefore\n\\begin{eqnarray}\n{\\rm l.h.s.}\n&=&[[L^l({\\alpha}), L^m_-(\\beta)]_+, L(\\gamma)]\n+[[L^l_+({\\alpha}), L^m_+(\\beta)], L(\\gamma)]+\n[L^m_+(\\beta), [L^l_+({\\alpha}), L(\\gamma)]]\\nonumber\\\\\n&=&[[L^l({\\alpha}), L^m_-(\\beta)]_+, L(\\gamma)]\n+[L^l_+({\\alpha}), [L^m_+(\\beta), L(\\gamma)]]\\nonumber\\\\\n&=&{\\rm r.h.s.}\\nonumber\n\\end{eqnarray}\nAll the other cases can be done in the same way. Therefore eqs.(4.2)\nreally define an integrable system.\nThe associated linear system is\n\\begin{eqnarray}\n\\left\\{\\begin{array}{l}\nL(1)\\Psi={\\lambda}\\Psi,\\\\\\noalign{\\vskip10pt}\n\\ddt {1,r}\\Psi=L^r_+(1)\\Psi,\\\\\\noalign{\\vskip10pt}\n\\ddt {{\\alpha},r}\\Psi=-L^r_-({\\alpha})\\Psi,\\qquad {\\alpha}=2,3,\\ldots,n,\\\\\\noalign{\\vskip10pt}\nM\\Psi={\\partial \\over {\\partial \\lm}}\\Psi.\n\\end{array}\\right.\\label{lmpsigen}\n\\end{eqnarray}\nIn fact we can rewrite this\nlinear system in a better way by choosing a new function\n\\begin{eqnarray}\n\\Psi({\\lambda},t)\\Longrightarrow\\xi({\\lambda},t)=\n\\exp(-\\sum_{r=1}^{\\infty}t_{1,r}{\\lambda}^r_1)\\Psi({\\lambda},t),\\nonumber\n\\end{eqnarray}\nthen all the flows can be\nsummarized by a single equation\n\\begin{eqnarray}\n\\ddt {{\\alpha}, r}\\xi=-L^r_-({\\alpha})\\xi.\n\\end{eqnarray}\nThe consistency conditions of the above linear system exactly give the\nhierarchy (4.2) We would like to remark\nthat the hierarchy (4.2) have several important\nsub--hierarchies.\n\n$(i)$. ${\\alpha}=\\beta=1$, the eqs.(\\ref{flowb}) are nothing but the\nusual { KP hierarchy } (\\ref{KPequation}).\n\n$(ii)$. $2\\leq{\\alpha}=\\beta\\leq  n$, the eqs.(\\ref{flowb}) give $(n-1)$\nKP hierarchies whose KP operator possess the form (\\ref{loperator}).\n\n$(iii)$. All the  flows commute.\n\n$(iv)$. When $n\\longrightarrow\\infty$, the full hiearchy (4.2)\ncontains all possible combinations of additional flows\n(\\ref{additional}).\n\nWe may  conclude that  the hierarchy (4.2)\npossess $n$ bi--hamiltonian structures, each  of  them generates a { KP hierarchy },\nall of these hierarchies couple together. The integrability of the system\nis guaranteed by the commutativity of the flows. However, we are not sure\nwhether the  hamiltonians in\ndifferent series are commutative.\n\n\\subsection{New bi--hamiltonian structures}\n\nIn the above analysis, all the operators are expanded in terms of ${\\partial}$.\nHowever,\nif we  use the same trick as the one in previous section,\nit is not difficult to reexpress them in terms of any one of $\\ddt {{\\alpha},1}$'s.\nLet us define\n\\begin{eqnarray}\n{\\partial}_{{\\alpha}}\\equiv {{\\partial}\\over{{\\partial} t_{{\\alpha},1}}}\n\\end{eqnarray}\nand expand $L_-({\\alpha})$ in powers of ${\\partial}$\n\\begin{eqnarray}\nL_-({\\alpha})=-\\sum_{i=1}^{\\infty}\\Gamma^{({\\alpha})}_i{\\partial}^{-i}\n\\end{eqnarray}\nthen the first flows of the linear system (\\ref{lmpsigen}) suggest\n\\begin{eqnarray}\n{\\partial}_{{\\alpha}}=\\sum_{i=1}^{\\infty}\\Gamma^{({\\alpha})}_i{\\partial}^{-i}\n\\end{eqnarray}\nsimilar to the argument in the previous section, we can invert these\nrelations, such that\n\\begin{eqnarray}\n{\\partial}=\\sum_{i=1}^{\\infty}{\\tilde\\Gamma}^{({\\alpha})}_i{\\partial}^{-i}_{{\\alpha}}\n\\end{eqnarray}\nSubstituting them into the formulas (\\ref{loperator}), we get the expansions\nof $L({\\alpha})$ in $\\ddt {\\beta,1}$ ({\\it for any ${\\alpha},\\beta$}). In particular\n$L({\\alpha})$ expanded in $\\ddt {{\\alpha},1}$ is also a KP operator,\n\\begin{eqnarray}\nL({\\alpha})=-({\\partial}_{{\\alpha}}+\\sum_{i=1}^{\\infty}v^{({\\alpha})}_i{\\partial}^{-i}_{{\\alpha}})\n\\end{eqnarray}\nand its ${\\alpha}-th$ series of flows is nothing but the ordinary KP hierarchy\n\\begin{eqnarray}\n\\ddt {{\\alpha}, r} L(\\alpha)=(-1)^{r+1}[L^r_+(\\alpha), L({\\alpha})]\n\\end{eqnarray}\nwhere the operators are expanded in powers of ${\\partial}_{{\\alpha}}$, and the additional\nsign indicates rescaling of the parameters. Of course,\nfor this subsystem, we can\nconstruct its integrable structure, by replacing $L$ in (\\ref{poisson1})\nand (\\ref{poisson2}) by $L({\\alpha})$.\nTherefore, we may say\nthat KP system (4.2) possesses multi bi--hamiltonian structures,\nand it contains ``$n$\" coupled ordinary KP hierarchies.\nThe coupling comes from the dynamical equations (4.2)\nwith ${\\alpha}\\neq\\beta$.\n\n\\subsection{The $\\tau$--function of the generalized { KP hierarchy }}\n\nUsing eqs.(4.2), we get\n\\begin{eqnarray}\n\\ddt {\\beta,s} \\rm res_{{\\partial}}L^r({\\alpha})\n=\\ddt {{\\alpha},r} \\rm res_{{\\partial}}L^s(\\beta),\\qquad \\forall {\\alpha},\\beta;\\quad r,s.\n\\end{eqnarray}\nThese equalities imply the existence of $\\tau$--function\n\\begin{eqnarray}\n{{\\partial}^2\\over{{\\partial} t_{1,1}{\\partial} t_{{\\alpha},r}}}\\ln\\tau=\\rm res_{{\\partial}}L^r({\\alpha}),\\qquad\n\\forall {\\alpha},r.\\label{taumulti}\n\\end{eqnarray}\nUsing this $\\tau$--function, we can introduce a series of the Baker--Akhiezer\nfunctions,\n\\begin{eqnarray}\n\\Psi_{{\\alpha}}(t,{\\lambda}_{{\\alpha}})={{\\tau(t_{{\\alpha},1}-{1\\over {{\\lambda}_{{\\alpha}}}},t_{{\\alpha},2}-\n{1\\over {2{\\lambda}^2_{{\\alpha}}}},\n\\ldots)}\\over{\\tau(t)}}e^{\\xi(t,{\\lambda}_{{\\alpha}})}\n\\end{eqnarray}\nwhere ${\\alpha}=1,2,\\ldots, n$. To each $\\Psi_{{\\alpha}}$ we can associate a linear\nsystem. Among them, the ${\\alpha}=1$ case was discribed above. The other\ncases can be analysed in the similar way\\cite{Thesis}.\n\n\\section{Discussion}\n\nWe have shown that the KP hierarchy can be extended to a much larger hierarchy\nby introducing additional KP operators. This generalized hierarchy can be\nconsidered as several coupled KP hierarchies. For each\nof the KP operators, we have constructed its  bi--hamiltonian structure\nby introducing new basic derivatives. Although we do not\nknow if all these hamiltonians are in involution,\nThe commutativity of the flows guarantees the integrability of the system.\n\nAs we know in the ordinary KP hierarchy case, the series of flows reflects the\nlarge symmetry of the system generated  by its Hamiltonians. In our case,\nthe multi--series of flows imply that this new hierarchy (4.2)\nshould possess a much larger symmetry.\nHowever we are not sure what this large symmetry is.\n\nIt is not clear if this new hierarchy relates to the multi--component\nKP hierarchy. Another interesting problem is to reduce this hierarchy to the\nknown hierarchies like generalized KdV hierarchies. This is under\ninvestigation.\n\n\\vskip0.8cm\n\\noindent\n{\\bf Achnowledgement}\n\n\\vskip 0.2cm\n\nI would like to thank Prof. L. Bonora for his constant encouragement,\nvaluable suggestions and fruitful discussions.\n\n\n\\renewcommand{\\Large}{\\normalsize} ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nLet ${\\mathcal O}$ be a discrete valuation ring with field of fractions $K$.\nTropicalization is a procedure which takes as input a $d$-dimensional subvariety of an algebraic torus over $K$,\n$X\\subset (K^*)^n$, and associates to it a balanced weighted rational \n$d$-dimensional polyhedral complex, $\\operatorname{Trop}(X)\\subseteq{\\mathbb R}^n$.  \nSeveral questions naturally arise in this framework; for instance, one may ask what combinatorial properties\nof $\\operatorname{Trop}(X)$ correspond to geometric properties of $X$.\nOne may also ask what constraints being a tropicalization places on the topology of a polyhedral complex.  In \n\\cite{Hacking}, Hacking proved that if $X$ is a subvariety of $({\\mathbb C}^*)^n$ satisfying a certain \ngenericity condition, then the link of the fan $\\operatorname{Trop}(X)$ only has reduced rational homology in \nthe top dimension.  Hacking's result holds for a number of examples, including generic \nintersections of ample hypersurfaces in projective toric varieties.  In \\cite[Sec. 10]{Sp3}, \nSpeyer showed that if $C$ is a genus $g$ curve in $(K^*)^n$ satisfying a genericity condition \nthen there exists a balanced metric graph $\\Gamma$ with $b_1(\\Gamma)\\leq g$ and a parameterization \n$i:\\Gamma\\rightarrow \\operatorname{Trop}(C)$ that is affine-linear on edges.  Our results include an \nanalogue of Hacking's result for varieties defined over $K$ or as a higher-dimensional \ngeneralization of Speyer's result.  One may consider the monodromy action of $\\operatorname{Gal}(K^{\\operatorname{sep}}\/K)$\non the \\'{e}tale cohomology $H^*_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}}, {\\mathbb Q}_l)$ and ask what properties of the monodromy action are encoded in $\\operatorname{Trop}(X)$.  This is the algebraic analogue of the monodromy action of a family of varieties defined over a punctured disk.  The work of the second-named author with Hannah Markwig and Thomas Markwig \\cite{KMM} relating the valuation of the $j$-invariant of a plane elliptic curve to the length of the cycle in its tropicalization can be seen in this light.  We give a generalization of that result. \n\n\n\nAll of our results require that the variety $X$ be {\\em sch\\\"{o}n}, a natural condition introduced \nin \\cite{Tevelev} and generalized by \\cite{LQ} to the nonconstant coefficient case.\nThis condition means that the ambient torus $(K^*)^n$ may be compactified to a \ntoric scheme ${\\mathbb P}$ over an extension of $\\operatorname{Spec} {\\mathcal O}$ such that the intersection of ${\\mathcal X}=\\overline{X}\\subset {\\mathbb P}$ \nwith each open torus orbit $U_P$ is smooth of the expected dimension.  For appropriate ${\\mathbb P}$, ${\\mathcal X}$ is then a simple normal\ncrossings degenerations of $X$ (c.f.~\\ref{prop:degeneration}).   The construction of the desired toric scheme ${\\mathbb P}$ follows from Proposition \\ref{prop:nc}, a technical result in polyhedral geometry which has not appeared, to our knowledge, in the literature before.\n\nThe existence of such simple normal crossings degenerations for a sch\\\"on $X$ allows us to construct \na natural ``parameterizing space'' $\\Gamma_X$.  This generalizes a construction introduced by\nSpeyer \\cite{Sp3} when $X$ has dimension $1$.  The space $\\Gamma_X$ is closely related to the\ndual complex of an appropriate degeneration ${\\mathcal X}$ of $X$; in this guise it already appears implicitly\nin \\cite{Hacking}, as well as in \\cite{HKT}.  Kontsevich-Soibelman use rigid analytic techniques\nto construct a similar polyhedral complex, with an integral affine structure,\nfrom a suitable degeneration of $X$ in~\\cite{KS}.\n\nThe space $\\Gamma_X$ we construct is {\\em independent} of a choice of model ${\\mathcal X}$ for $X$;\nit depends only on $X$ and its embedding in the torus.  Moreover, $\\Gamma_X$ comes equipped with a canonical\nmap to $\\operatorname{Trop}(X)$.  A choice of sufficiently fine triangulation of $\\operatorname{Trop}(X)$ gives \n$\\Gamma_X$ the structure of a polyhedral complex.  When $\\Gamma_X$ is viewed in such a way,\nthe natural parameterization $\\Gamma_X\\rightarrow \\operatorname{Trop}(X)$ is affine-linear on polyhedra.  \nThis parameterization has several nice properties.\nFor instance, it is natural under monomial morphisms: if $X$ and $Y$ are sch\\\"{o}n \nsubvarieties of tori $T$ and $T'$ and $\\phi:T\\rightarrow T'$ is a homomorphism taking $X$ to $Y$, then \nthere is an induced map of complexes $\\Gamma_X\\rightarrow\\Gamma_Y$ that commutes with parameterizations.  \nMoreover, $\\Gamma_X$ satisfies a balancing condition analogous to the one satisfied by all \ntropical varieties.  Finally, it is ``not far'' from $\\operatorname{Trop}(X)$: if the intersections of ${\\mathcal X}$\nwith open torus orbits $U_P$ in ${\\mathbb P}$ satisfy certain connectedness hypotheses,\nwe may equate the cohomology of $\\Gamma_X$ and $\\operatorname{Trop}(X)$ in certain \ndegrees.  We hope that parameterizing complexes will be seen as a \nfundamental object in tropical geometry.\n\nOur main results (principally Theorem~\\ref{thm:main}, Corollary~\\ref{cor:CI}, and Proposition\\ref{prop:npower}) relate\nthe cohomology of $\\Gamma_X$ to geometric invariants of $X$.  In particular we consider the\n{\\'e}tale cohomology $H^*_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}}, {\\mathbb Q}_l)$; this cohomology comes equipped with a\nnatural filtration, called the weight filtration.  We construct a natural isomorphism between the\n``weight $0$'' subquotient $W_0 H^r_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}}, {\\mathbb Q}_l)$ arising from this filtration\nand the cohomology $H^r(\\Gamma_X, {\\mathbb Q}_l)$ of $\\Gamma_X$.  We then use this to show\nthat if $X$ is the generic intersection of ample hypersurfaces in a toric scheme ${\\mathbb P}$, \nthen $H^r(\\operatorname{Trop}(X),{\\mathbb Q}_l)$ vanishes for $1\\leq r<\\dim X$, a non-constant coefficient analogue of \nHacking's result.  We also prove results about the monodromy action on the middle cohomology $H^n_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}}, {\\mathbb Q}_l)$.\n\nThe main tool we use is the Rapoport-Zink weight spectral sequence \\cite{RZ}.  Under the sch\\\"{o}n \ncondition, after a finite base-extension ${\\mathcal O}^{\\prime}$ of ${\\mathcal O}$, we may compactify the ambient torus to \na toric scheme ${\\mathbb P}$ defined over ${\\mathcal O}^{\\prime}$ so that the central fiber of the closure\n${\\mathcal X}$ of $X$ in ${\\mathbb P}$ is a divisor with simple normal crossings.  The divisor, a degeneration of $X$, \nhas a stratification coming from intersections of its irreducible components.  The Rapoport-Zink\nspectral sequence then gives a very explicit formula for the cohomology on $X$, together with its\nweight filtration, in terms of these strata.  The $E_1$-term \nof the weight spectral sequence is formed from the cohomology groups of the strata with \nboundary maps built from the data of restriction maps and Gysin maps.  The spectral sequence \nconverges to the cohomology of the general fiber, and the induced filtration is the weight filtration.  \nMoreover, the weight spectral sequence \ndegenerates at $E_2$.  We thus obtain an explicit description of the smallest nontrivial piece of \nthe filtration which is isomorphic to the cohomology groups of $\\Gamma_X$.\n\nIt is interesting to compare this result to results of Berkovich \\cite{B} on rigid analytic spaces.\nIn particular, Berkovich shows that the cohomology group \n$H^r_{\\mbox{\\rm \\tiny \\'et}}(\\overline{X}_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{-r}$ arising in our result is isomorphic to\nthe cohomology of the Berkovich space $X^{\\mbox{\\rm \\tiny an}}$ attached to $X$.  Our result thus suggests \na strong link between $\\Gamma_X$ and $X^{\\mbox{\\rm \\tiny an}}$.  In fact, Speyer \\cite{Speyer} constructs a\nnatural map from $X^{\\mbox{\\rm \\tiny an}}$ to $\\operatorname{Trop}(X)$.  This map factors through the map $\\Gamma_X \\rightarrow \\operatorname{Trop}(X)$,\nand it is natural to ask if the resulting map $X^{\\mbox{\\rm \\tiny an}} \\rightarrow \\Gamma_X$ map is a homotopy \nequivalence.  Links between tropical geometry and rigid\ngeometry have also appeared in works of Einsiedler-Kapranov-Lind~\\cite{EKL}, and Payne~\\cite{P}.\n\nUnder additional hypotheses, one can relate the results above to questions involving monodromy.\nA variety defined over $\\operatorname{Spec} K$ is analogous to a family of varieties defined \nover a punctured disk.  \nThe fundamental group of the punctured disk acts on the cohomology of a general fiber of such a\nfamily by monodromy.  The analogue of this monodromy action for varieties over $\\operatorname{Spec} K$ is the\naction of the inertia group $I_K$ of $K$ on the \\'{e}tale cohomology \n$H^*_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)$.  After a \npossible finite base-extension of ${\\mathcal O}$, this action is unipotent, and is \ngiven by the {\\em monodromy operator}\n$$N: H^i_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l) \\rightarrow H^i_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l(-1))$$\nan endomorphism of the \\'etale cohomology that is essentially the (matrix) logarithm of\nthe action.  (We refer the reader to section~\\ref{sec:monodromy} for precise\ndefinitions).  The action of $N$ induces an increasing filtration on the cohomology.\nThe weight-monodromy conjecture asserts that this filtration coincides (up to a shift in degree) \nwith the weight filtration\ndescribed above.  Although it is not completely settled, this conjecture is known to be\ntrue in many cases of interest; for instance, it is known $X$ is a curve, surface, or\nan abelian variety.  Ito \\cite{Ito} has proven the weight-mondromy conjecture when ${\\mathcal O}$ has\nequal characteristic.  Thus, in these situations, one can interpret Theorem~\\ref{thm:main}\nas an isomorphism between the cohomology of $\\Gamma_X$ and the smallest\nnontrivial piece of the monodromy filtration of the cohomology of $\\overline{X}_K$, the closure \nof $X_K$ in the generic fiber of ${\\mathbb P}$:\n$$H^r(\\Gamma_X, {\\mathbb Q}_l) \\cong H^r_{\\mbox{\\rm \\tiny \\'et}}(\\overline{X}_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{-r}.$$\nAs a consequence, Corollary \\ref{cor:bound} bounds \nthe Betti numbers of $\\Gamma_X$ in terms of those of $X$:\n$$b_r(\\Gamma_X) \\leq \\frac{1}{r+1} b_r(X)$$\ngeneralization Speyer's result on curves.\n  \nWe apply our techniques to get a description of some of the monodromy action and not just of the monodromy filtration.  In Proposition \\ref{prop:npower}, we give an interpretation of the top power of monodromy operator acting on the middle-dimensional cohomology\n\\[N^d:H^d_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{d}\/H^d_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{d-1}\\rightarrow H^d_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{-d}(-d)\\]\nusing the isomorphisms \n\\begin{eqnarray*}\nH^d_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{d}\/H^d_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{d-1}&\\cong&H_d(\\Gamma,{\\mathbb Q}_l)(-d)\\\\ \nH^d_{\\mbox{\\rm \\tiny \\'et}}(\\overline{X}_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{-d}&\\cong&H^d(\\Gamma_X, {\\mathbb Q}_l).\n\\end{eqnarray*}\nThe operator can be viewed as a bilinear pairing on $H_d(\\Gamma_X, {\\mathbb Q}_l)$ in which case it is the volume pairing that takes a pair of top-dimensional cycles to the oriented volume of their intersection.  This specializes to the length pairing in the case of curves.  In the case of genus $1$ curves, by a straightforward application of the conductor-discriminant formula, we are able to recover the spatial sch\\\"on analogue of a result proved by the second-named author with H. Markwig andT. Markwig \\cite{KMM}: the valuation of the $j$-invariant of an elliptic curve $X$ with potentially multiplicative reduction is equal to $-a$ where $a$ is the length of the unique cycle in $\\Gamma_X$.  \n\nOur techniques are very similar to those of Hacking and Speyer.  Hacking uses a spectral sequence coming \nfrom a weight filtration on a complex of differential forms while we use the Rapoport-Zink spectral \nsequence.  Speyer's results use a resolution of the structure sheaf of a degeneration of a curve $C$ coming \nfrom a stratification induced by a toric scheme while we use a locally constant sheaf.\n\nWe should mention some related results.  In \\cite{GS}, Gross and Siebert construct \na scheme $X_0$ from an integer affine manifold and a toric polyhedral decomposition.  If $X_0$ is \nembedded in a family ${\\mathcal X}$ over ${\\mathbb C}[[t]]$, they are able to determine the limit mixed Hodge structure \nin terms of the combinatorial data.  In \\cite{TMNF}, the second-named author and Stapledon give a description of the Hodge-Deligne polynomial of the limit mixed Hodge structure of a sch\\\"{o}n family of varieties over a punctured disk in terms of $\\operatorname{Trop}(X)$ and initial degenerations of $X$.\n\nWe would like to thank Brian Conrad, Richard Hain, Kalle Karu, Sean Keel, Sam Payne, Zhenhua Qu, Bernd Siebert, Martin Sombra, \nDavid Speyer, Alan Stapledon, and Bernd Sturmfels for valuable discussions.\n\n\\section{Toric Schemes}\n\nWe begin by reviewing a construction that attaches a degenerating family of\ntoric varieties over a discrete valuation ring to a rational polyhedral complex in ${\\mathbb R}^n$.\nThis has appeared several times in the literature~\\cite{Speyer}~\\cite{NS}\n~\\cite{Smirnov}.  We follow the approach of~\\cite{NS} here.  Fix a discrete\nvaluation ring ${\\mathcal O}$, with field of fractions $K$ and residue field $k$, and \na uniformizer $\\pi$ of ${\\mathcal O}$.\n\n\\begin{defn} A rational polyhedral complex in ${\\mathbb R}^n$ is a collection \n$\\Sigma$ of finitely many convex rational polyhedra\n$P \\subset {\\mathbb R}^n$ with the following properties:\n\\begin{itemize}\n\\item If $P \\in \\Sigma$ and $P^{\\prime}$ is a face of $P$, then $P^{\\prime}$\nis in $\\Sigma$.\n\\item If $P,P^{\\prime} \\in \\Sigma$ then $P \\cap P^{\\prime}$ is a face of both\n$P$ and $P^{\\prime}$.\n\\end{itemize}\nIf, in addition, the union $\\bigcup_{P\\in\\Xi} P$  is equal to ${\\mathbb R}^n$, then $\\Sigma$ is said to be {\\em complete}.\n\\end{defn}\n\n\nGiven a $\\Sigma$ as above, we can construct a fan ${\\tilde \\Sigma}$ in \n${\\mathbb R}^n \\times {\\mathbb R}_{\\geq 0}$ as follows: for each $P \\in \\Sigma$ let\n${\\tilde P}$ be the closure in ${\\mathbb R}^n \\times {\\mathbb R}_{\\geq 0}$ of the set\n$$\\{(x,a) \\subset {\\mathbb R}^n \\times {\\mathbb R}_{>0} : \\frac{x}{a} \\in P\\}.$$\nThen ${\\tilde P}$ is a rational polyhedral cone in ${\\mathbb R}^n \\times {\\mathbb R}_{>0}$.  Its \nfacets come in two types:\n\\begin{itemize}\n\\item cones of the form ${\\tilde P}^{\\prime}$, where $P^{\\prime}$ is a facet of $P$, and\n\\item the cone $P_0 = {\\tilde P} \\cap ({\\mathbb R}^n \\times \\{0\\})$, which is the limit as $a$ goes to zero\nof the polyhedron $aP$ in ${\\mathbb R}^n$.\n\\end{itemize}\nWe let ${\\tilde \\Sigma}$ be the collection of cones of the form ${\\tilde P}$ and $P_0$ for $P$ in $\\Sigma$.\nIf $\\Sigma$ is a subcomplex of a complete rational polyhedral complex, we say that $\\Sigma$ is {\\em completable} and, by Corollary 3.12 of \\cite{BGS}, ${\\tilde \\Sigma}$ is a rational polyhedral fan in ${\\mathbb R}^n \\times {\\mathbb R}_{\\geq 0}$.   Note that\n$\\Sigma = {\\tilde \\Sigma} \\cap ({\\mathbb R}^n \\times \\{1\\})$.  On the other hand\nthe fan $\\Sigma_0$ given by ${\\tilde \\Sigma} \\cap ({\\mathbb R}^n \\times \\{0\\})$ is the limit as\n$a$ approaches zero of the polyhedral complexes $a\\Sigma$.\n\n\\begin{rem} \\rm In fact, by results of \\cite{BGS}, the association $\\Sigma \\mapsto {\\tilde \\Sigma}$ defines a bijection\nbetween the set of complete polyhedral complexes in ${\\mathbb R}^n$ and the set of complete fans in \n${\\mathbb R}^n \\times {\\mathbb R}_{\\geq 0}$ for which every cone contained in ${\\mathbb R}^n \\times \\{0\\}$\nis the boundary of a cone that meets ${\\mathbb R}^N \\times {\\mathbb R}_{>0}$.  Its inverse is\n${\\tilde \\Sigma} \\mapsto {\\tilde \\Sigma} \\cap ({\\mathbb R}^n \\times \\{1\\}).$\n\\end{rem}\n\nLet $X({\\tilde \\Sigma})_{{\\mathbb Z}}$ be the toric scheme over ${\\mathbb Z}$ associated to the fan ${\\tilde \\Sigma}$.  (The\nconstruction associating a toric variety to a fan is usually given over a field, but works\njust as well with coefficients in ${\\mathbb Z}$.)\nProjection from ${\\mathbb R}^n \\times {\\mathbb R}_{\\geq 0}$ to ${\\mathbb R}_{\\geq 0}$ induces a map of \nfans from ${\\tilde \\Sigma}$ to the fan\n$\\{0,{\\mathbb R}_{\\geq 0}\\}$ associated to ${\\mathbb A}^1_{{\\mathbb Z}}$.  This gives rise to a map of\ntoric varieties $\\pi_{{\\mathbb Z}}: X({\\tilde \\Sigma})_{{\\mathbb Z}} \\rightarrow {\\mathbb A}^1_{{\\mathbb Z}}$.  As remarked in~\\cite{NS},\nthis map is flat and torus equivariant.  Let $\\iota: \\operatorname{Spec} {\\mathcal O} \\rightarrow {\\mathbb A}^1_{{\\mathbb Z}}$\nbe the map corresponding to the map ${\\mathbb Z}[t] \\rightarrow {\\mathcal O}$ that takes $t$ to $\\pi$. \nWe let $X({\\tilde \\Sigma})$ be the scheme over ${\\mathcal O}$ obtained by base change from $X({\\tilde \\Sigma})_{{\\mathbb Z}}$\nvia $\\iota$, and let $\\pi: X({\\tilde \\Sigma}) \\rightarrow {\\mathcal O}$ be the projection.\n\nWe summarize results of~\\cite{NS} concerning this construction:\n\\begin{itemize}\n\\item The general fiber $X({\\tilde \\Sigma}) \\times_{\\operatorname{Spec} {\\mathcal O}} \\operatorname{Spec} K$ is isomorphic\nto the toric variety over $K$ associated to $\\Sigma_0$.\n\\item If $\\Sigma$ is {\\em integral}, i.e. the vertices of every polyhedron in $\\Sigma$\nlie in ${\\mathbb Z}^n$, then the special fiber $X({\\tilde \\Sigma})_k = X({\\tilde \\Sigma}) \\times_{\\operatorname{Spec} {\\mathcal O}} \\operatorname{Spec} k$\nis reduced.   \n\\item There is an inclusion-reversing bijection\nbetween closed torus orbits in $X({\\tilde \\Sigma})_k$ and polyhedra $P$ in $\\Sigma$; the irreducible \ncomponents of $X({\\tilde \\Sigma})_k$ correspond to vertices in $\\Sigma$; the intersection of a collection\nof irreducible components corresponds to the smallest polyhedron in $\\Sigma$ containing all\nof their vertices.\n\\end{itemize}\n\nNote that adjoining a $d$th root of $\\pi$ to ${\\mathcal O}$ has the effect\nof rescaling $\\Sigma$ by $d$; that is, if ${\\mathcal O}^{\\prime} = {\\mathcal O}[\\pi^{\\frac{1}{d}}]$, then\nthe base change of the family $X({\\tilde \\Sigma}) \\rightarrow {\\mathcal O}$ is the family \n$X(\\widetilde{(d\\Sigma)}) \\rightarrow {\\mathcal O}^{\\prime}$.  \nIn particular,\ngiven any toric scheme coming from a polyhedral complex $\\Sigma$, we can choose $d$\nsuch that $d\\Sigma$ is integral; after taking a suitable ramified base change of ${\\mathcal O}$\nthe special fiber of the family $X({\\tilde \\Sigma}) \\rightarrow {\\mathcal O}$ will be reduced.\n\nWe will be particularly interested in degenerations of toric varieties in which the special\nfiber is a divisor with simple normal crossings.  These are easy to construct, because the\nboundary of a smooth toric variety is always a divisor with simple normal crossings.  We make use of the following result in polyhedral combinatorics:\n\n\\begin{prop} \\label{prop:nc}\nLet $\\Sigma$ be a complete rational polyhedral complex in ${\\mathbb R}^n$. \nThere exists an integer $d$, and a subdivision $\\Sigma^{\\prime}$ of $d\\Sigma$ such that the general\nfiber of the scheme $X({\\tilde \\Sigma}^{\\prime})$ is a smooth toric variety\nand the special fiber of $X({\\tilde \\Sigma}^{\\prime})$ is a divisor with simple normal crossings.  Moreover, if the recession fan $\\Sigma_0$ is already simplicial and unimodular, $\\Sigma^{\\prime}$ can be chosen to have $\\Sigma_0'=\\Sigma_0$.\n\\end{prop}\n\\begin{proof} Choose an integer $l_1$ such that $l_1\\Sigma$ is integral.\nFulton~\\cite[Sec. 2.6]{Fulton} gives an algorithm for constructing a subdivision\n${\\tilde \\Sigma}^{\\prime}$ of the fan $\\widetilde{l_1\\Sigma}$ such that all the cones of ${\\tilde \\Sigma}_1$\nare simplicial and unimodular.  Pick an integer $l_2$ sufficiently divisible so $\\Sigma_1={\\tilde \\Sigma}^{\\prime} \\cap ({\\mathbb R}^n \\times \\{l_2\\})$ is integral.  $\\Sigma_1$ is a subdivision of ${\\mathbb R}^n$ with the property that each of its recession cones is simplicial.  Because ${\\tilde \\Sigma}_1$ is simplicial each of its cones is of the form $\\tilde{P}+Q_0$ where $P$ is a \nsimplex in $\\Sigma_1$, $Q_0$ is a cone in the recession fan $(\\Sigma_1)_0$, and $\\tilde{P}\\cap Q_0=\\{0\\}$.    Consequently, the corresponding polyhedron of $\\Sigma_1$ is $(\\tilde{P}+Q_0)\\cap ({\\mathbb R}^n\\times\\{l_2\\})=P+Q_0$.\nFor a polyhedron $F$ in ${\\mathbb R}^k$, let $N_F={\\mathbb Z}^k\\cap {\\operatorname{Span}}_{\\mathbb R}(F-w)$ \nwhere $w$ is a point of $F$.  If $F$ is a rational polytope, then $N_F$ has the property a basis of it can be extended to a basis of ${\\mathbb Z}^k$.\n\nWe claim $N_P+N_{Q_0}=N_{P+Q_0}$ for every cone $\\tilde{P}+Q_0$ of ${\\tilde \\Sigma}_1$.  It is clear that $N_P+N_{Q_0}\\subseteq N_{P+Q_0}$.  The inclusion ${\\mathbb Z}^n\\hookrightarrow{\\mathbb Z}^{n+1}={\\mathbb Z}^n\\times{\\mathbb Z}$ given by $x\\mapsto (x,l_2)$ identifies $N_{P+Q_0}$ with the intersection of $N_{\\tilde{P}+Q_0}$ with ${\\mathbb Z}^n\\times \\{l_2\\}$.   Since $\\tilde{P}+Q_0$ is unimodular, \n$N_{\\tilde{P}}+N_{Q_0}$ is equal to $N_{\\tilde{P}+Q_0}$.    Therefore, if $x\\in N_{P+Q_0}$, $(x,l_2)=x_{\\tilde{P}}+x_{Q_0}$ where $x_{\\tilde{P}}\\in \\tilde{P}$ and $x_{Q_0}\\in Q_0$.  Since the last coordinate of $x_{Q_0}$ is $0$, $x_{\\tilde{P}}=(x_P,l_2)$ for $x_P\\in N_P$.  It follows that $x=x_P+x_{Q_0}$.\n\nLet $\\Sigma_1^b$ be the union of the bounded polyhedra of $\\Sigma_1$. \nBy an important step of the proof of semi-stable reduction \\cite[Ch. 3, Thm 4.1]{KKMS}, there is an integer $l_2$ and a unimodular triangulation ${\\Sigma_1^b}'$ of $l_2\\Sigma_1^b$.  This induces a subdivision of $l_2\\Sigma_1$ where the polyhedra whose relative interior is contained in the relative interior of $l_2(P+Q_0)$  are of the form $P'+Q_0$ where $P'$ is a simplex in ${\\Sigma_1^b}'$ whose relative interior is contained in the the relative interior of $l_2P$.  We call this subdivision $\\Sigma'$.  It is simplicial by construction.  We claim that it is also unimodular.   It suffices to show that maximal cones in ${\\tilde \\Sigma}'$ are unimodular.  Let $\\tilde{P}'+Q_0$ be a  maximal cone in ${\\tilde \\Sigma}'$.  Then the relative interior of $P'$ is contained in the relative interior of $l_2P$ with $\\dim P=\\dim P'$.   Since $P'$ is unimodular, its ${\\mathbb Z}$-affine span is $N_{P'}$.    Consequently, since $N_{P'}+N_{Q_0}=N_P+N_{Q_0}=N_{P+Q_0}$, we see that any element of $N_{P+Q_0}$ can be written as integer combination $\\sum m_i v_i+\\sum n_iw_i$ where $v_i$ are vertices of $P'$, $w_i$ are the primitive vectors along the rays of $Q_0$, and $\\sum m_i=1$.  Consequently, any element of $N_{P+Q_0}\\times \\{1\\}$ can be written as  an integer combination of the primitive vectors along the rays of $\\tilde{P'}+Q_0$.  Consequently these vectors generate $N_{\\tilde{P}+Q_0}\\subset{\\mathbb Z}^{n+1}$.  Therefore $\\tilde{P}'+Q_0$ is smooth.\n\nIf $\\Sigma_0$ was simplicial and unimodular to begin with, none of these steps would have affected the cones $Q_0$ of $\\Sigma_0$\n\nThe upshot is that $X({\\tilde \\Sigma}^{\\prime})$ is a smooth toric variety\nwith a birational morphism $X({\\tilde \\Sigma}^{\\prime}) \\rightarrow X({\\tilde \\Sigma})$.  The induced map\n$X({\\tilde \\Sigma}^{\\prime}) \\rightarrow {\\mathcal O}$ is the toric scheme associated to\nthe integral polyhedral complex $\\Sigma^{\\prime}$.\nThe general fiber of $X({\\tilde \\Sigma}^{\\prime})$ over ${\\mathcal O}$ corresponds to the fan \n$\\Sigma^{\\prime}_0$, and is therefore smooth.  The special fiber is a union of\nirreducible components of the boundary of $X({\\tilde \\Sigma}^{\\prime})$ and is therefore\na divisor of $X({\\tilde \\Sigma}^{\\prime})$ with simple normal crossings.\n\\end{proof}\n\n\\section{Tropical Degenerations}\n\nWe now describe the applications of tropical geometry to the study of degenerations of varieties over $K$.\nThese techniques have their origins in the Speyer's thesis \\cite{Speyer}.  \nThe approach we take here is due to Tevelev \\cite{Tevelev} in the ``constant coefficient case''; \nthe extension of Tevelev's work to the case of an arbitrary DVR done by Luxton and Qu \\cite{LQ}.\n\nLet $\\overline{K}$ be an algebraic closure of $K$.  There is a unique valuation \n$$\\operatorname{ord}: \\overline{K} \\rightarrow {\\mathbb Q}$$\nsuch that $\\operatorname{ord}(\\pi) = 1$.  \n\nLet ${\\mathcal T} \\cong {\\mathbb G}_m^n$ be a split $n$-dimensional torus over ${\\mathcal O}$, and let\n$T = {\\mathcal T} \\times_{{\\mathcal O}} K$ be the corresponding torus over $K$.  \nThe valuation $\\operatorname{ord}$\ngives rise to a map\n$$\\operatorname{val}: {\\mathcal T}(\\overline{K}) \\rightarrow {\\mathbb Q}^n,$$\nby fixing an isomorphism of ${\\mathcal T}$ with ${\\mathbb G}_m^n$ (and hence an isomorphism of ${\\mathcal T}(\\overline{K})$ \nwith $(\\overline{K}^*)^n$.)\nLet $X$ be a closed subvariety of $T$, defined over $K$.\n\n\\begin{defn}[\\cite{EKL}, 1.2.1]: The tropical variety $\\operatorname{Trop}(X)$ associated to $X$ is\nthe closure of $\\operatorname{val}(X(\\overline{K}))$ in ${\\mathbb R}^n$.\n\\end{defn}\n\nGiven such an $X$, one can ask for a well-behaved compactification $\\overline{X}$ of $X$,\nand a well-behaved degeneration of $\\overline{X}$ over ${\\mathcal O}$.  The problem of\nfinding such a degeneration is intimately connected to the set $\\operatorname{Trop}(X)$.\n\nLet $\\Sigma$ be a completable rational polyhedral complex in ${\\mathbb R}^n$, and\nlet ${\\mathbb P}$ be the corresponding toric scheme over ${\\mathcal O}$.  Identify\nthe group of cocharacters of ${\\mathcal T}$ with ${\\mathbb Z}^n$ in ${\\mathbb R}^n$; this identifies ${\\mathbb T}$ with \nthe open torus orbit on ${\\mathbb P}$.\n\nWe can thus take the closure of ${\\mathcal X}$ of $X$ in ${\\mathbb P}$.  By \\cite{Speyer}, 2.4.1, \nthe scheme ${\\mathcal X}$ is proper over ${\\mathcal O}$ if, and only if, $\\operatorname{Supp} \\Sigma$ contains\n$\\operatorname{Trop}(X)$.  We assume henceforth that $\\operatorname{Supp} \\Sigma$ contains $\\operatorname{Trop}(X)$.  Let $\\overline{X}$ be the fiber\nof ${\\mathcal X}$ over $K$, and ${\\mathcal X}_k$ be the special fiber of ${\\mathcal X}$.  The natural multiplication map\n$${\\mathcal T} \\times_{{\\mathcal O}} {\\mathbb P} \\rightarrow {\\mathbb P}$$ \nrestricts to a multiplication map\n$$m: {\\mathcal T} \\times_{{\\mathcal O}} {\\mathcal X} \\rightarrow {\\mathbb P}.$$ \n\n\\begin{defn} The pair $(X,{\\mathbb P})$ is {\\em tropical} if the map\n$$m: {\\mathcal T} \\times_{{\\mathcal O}} {\\mathcal X} \\rightarrow {\\mathbb P}$$\nis faithfully flat, and ${\\mathcal X} \\rightarrow {\\mathcal O}$ is proper.\n\\end{defn}\n\nWe then have the following, due to Tevelev \\cite{Tevelev} in the constant coefficient case.\nIn the non-constant coefficient case they can be found in \\cite{LQ}.\n\n\\begin{prop} A subvariety $X\\subset(\\overline{K}^*)^n$ admits a tropical pair $(X,{\\mathbb P})$.  \n\\end{prop}\n\n\\begin{prop}\nSuppose $(X,{\\mathbb P})$ is tropical and let\n${\\mathbb P}^{\\prime} \\rightarrow {\\mathbb P}$ be a morphism of toric schemes corresponding\nto a refinement $\\Sigma^{\\prime}$ of $\\Sigma$.  Then $(X,{\\mathbb P}^{\\prime})$ is also tropical. \n\\end{prop}\n\n\\begin{prop} If $(X,{\\mathbb P})$ is a tropical pair then $\\operatorname{Supp} \\Sigma = \\operatorname{Trop}(X)$.\n\\end{prop}\n\nFollowing Speyer (\\cite{Speyer}, 2.4) If $(X,{\\mathbb P})$ is a tropical pair, we call $\\overline{X}$ a \n{\\em tropical compactification} of $X$, and ${\\mathcal X}_k$ a {\\em tropical degeneration} of $X$.\n\nThe combinatorics of the special fiber of a tropical degeneration of $X$ is closely\nrelated to the combinatorics of $\\operatorname{Trop}(X).$  In particular if $(X,{\\mathbb P})$ is a tropical\npair, and ${\\mathcal X}$ is the corresponding tropical degeneration, then a polyhedron $P$\nof $\\Sigma$ corresponds to the closure of a torus orbit in the special fiber of ${\\mathbb P}$.  \nCall this torus orbit closure ${\\mathbb P}_P$.  Then the intersection ${\\mathcal X}_P$ of ${\\mathcal X}$ with ${\\mathbb P}_P$ \nis a closed subscheme of ${\\mathcal X}_k$.  Moreover, if $P$ and $P^{\\prime}$ are\npolyhedra of $\\Sigma$, and $Q$ is the smallest polyhedron in $\\Sigma$\ncontaining both $P$ and $P^{\\prime}$, then the intersection of ${\\mathcal X}_P$ and ${\\mathcal X}_{P^{\\prime}}$\nis ${\\mathcal X}_Q$.\n\nLet $U_P$ be the open torus orbit corresponding to $P$.\nFix a point $p$ in $U_P$, and consider the fiber over $p$\nof the multiplication map\n$$m: {\\mathcal T} \\times_{{\\mathcal O}} {\\mathcal X} \\rightarrow {\\mathbb P}.$$\nOn the one hand, $m^{-1}(p)$ is nonempty of dimension equal to the dimension of $X$,\nsince $m$ is flat and surjective.  On the other hand, by projection onto ${\\mathcal X}$, $m^{-1}(p)$ \nis isomorphic to the product ${\\mathcal T}_P\\times ({\\mathcal X} \\cap U_P)$, where ${\\mathcal T}_P$ is the subgroup \nof ${\\mathcal T}_0$ that acts trivially on $U_P$.  Since $({\\mathcal X} \\cap U_P)$ is dense in ${\\mathcal X}_P$ we find that\n${\\mathcal X}_P$ is nonempty of dimension equal to the dimension of $X$ minus the dimension of $P$.\n\nOn the other hand, let $w$ be a point in the relative interior of $P$.  Then $w$ corresponds\nto a cocharacter of $T$, and $w(\\pi)$ specializes to a point $p$ in $U_P$.   Projection\nonto ${\\mathcal T}$ identifies $m^{-1}(p)$ with the mod $\\pi$ reduction $\\operatorname{in}_w X$ of $w(\\pi)X$.\n(More formally, $\\operatorname{in}_w X$ can be defined as the special fiber of the closure in ${\\mathcal T}$\nof the subscheme $w(\\pi)X$ of $T$.  Note that this depends only on $X$ and $T$, not on our choice of\n$\\Sigma$.) In particular we have \n$$\\operatorname{in}_w X \\cong {\\mathcal T}_P \\times ({\\mathcal X} \\cap U_P)$$ \nfor any $w$ in the relative interior of $P$.  We have thus shown:\n\n\\begin{lemma} \\label{lemma:comp}\nThe space $\\operatorname{in}_w X$ is a torus bundle over ${\\mathcal X} \\cap U_P$.  In particular, if\n$C(\\operatorname{in}_w X)$ is the set of connected components of $(\\operatorname{in}_w X)_{\\overline{k}}$,\nand $C({\\mathcal X} \\cap U_P)$ is the set of connected components of $({\\mathcal X} \\cap U_P)_{\\overline{k}}$,\nthen the maps\n$$\\operatorname{in}_w X \\cong m^{-1}(p) \\rightarrow {\\mathcal X} \\cap U_P$$  \ngive a natural bijection of $C(\\operatorname{in}_w X)$ with $C({\\mathcal X} \\cap U_P)$.\n\\end{lemma}\n\nWe will be particularly interested in tropical pairs $(X,{\\mathbb P})$ where the multiplication\nmap $m: {\\mathcal T} \\times_{{\\mathcal O}} {\\mathcal X} \\rightarrow {\\mathbb P}$ is {\\em smooth}.  This condition is\ndue to Tevelev.\n\n\\begin{defn} A subvariety $X$ of ${\\mathcal T}$ is {\\em sch\\\"on}\nif there exists a tropical pair $(X,{\\mathbb P})$ such that the multiplication map\n$$m: {\\mathcal T} \\times_{{\\mathcal O}} {\\mathcal X} \\rightarrow {\\mathbb P}$$ is smooth.\n\\end{defn}\n\nOne then has (\\cite{LQ}, 6.7):\n\\begin{prop} If $X$ is sch\\\"on, then for {\\em any} tropical pair\n$(X,{\\mathbb P})$, the multiplication map\n$$m: {\\mathcal T} \\times_{{\\mathcal O}} {\\mathcal X} \\rightarrow {\\mathbb P}$$ \nis smooth.\n\\end{prop}\n\nNote that if $X$ is sch\\\"on then it is smooth (consider the preimage of the identity in T\nunder the multiplication map.)  In fact, we have:\n\n\\begin{prop} \\label{prop:schon} The following are equivalent:\n\\begin{enumerate}\n\\item $X$ is sch\\\"on.\n\\item $\\operatorname{in}_w X$ is smooth for all $w \\in \\operatorname{Trop}(X)$.\n\\item For any tropical pair $(X,{\\mathbb P})$, and any polyhedron $P$ in $\\Sigma$,\n${\\mathcal X} \\cap U_P$ is smooth.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\nStatements 2) and 3) are clearly equivalent since we have seen that $\\operatorname{in}_w X$ is the product\nof a torus with ${\\mathcal X} \\cap U_P$, where $P$ is the polyhedron in $\\Sigma$ that contains $w$ in its\nrelative interior.\n\nAs for the equivalence of 1) and 2), fix a tropical pair $(X,{\\mathbb P})$.  We have seen that the \nfibers of the multiplication map\n$$m: {\\mathcal T} \\times_{{\\mathcal O}} {\\mathcal X} \\rightarrow {\\mathbb P}$$ \nare isomorphic to $\\operatorname{in}_w X$ as $w$ ranges over $\\operatorname{Trop}(X)$.  So 1) implies 2) is clear.\nFor the converse, note that since $m$ is faithfully\nflat, to show it is smooth it suffices to show that it has smooth fibers.  \n\\end{proof}\n\nIt is easy to construct tropical degenerations of sch\\\"on varieties $X$ in which the special\nfiber is a divisor with simple normal crossings.  In particular we have:\n\n\\begin{prop}[c.f.~\\cite{Hacking}, proof of 2.4] \\label{prop:degeneration}\nLet $X$ be sch\\\"on.  There exists an integer $d$ and a tropical pair $(X,{\\mathbb P})$ over \n${\\mathcal O}[\\pi^{\\frac{1}{d}}]$ such that\n$\\overline{X}$ is smooth over $K[\\pi^{\\frac{1}{d}}]$, and ${\\mathcal X}_k$ is a divisor in ${\\mathcal X}$ with \nsimple normal crossings.  \n\\end{prop}\n\\begin{proof}\nLet $(X,{\\mathbb P})$ be any tropical pair over ${\\mathcal O}$, and let $\\Sigma$ be the rational polyhedral\ncomplex corresponding to ${\\mathbb P}$.  By Proposition~\\ref{prop:nc}, we can\nfind a refinement $\\Sigma^{\\prime}$ of $\\Sigma$ and an integer $d$ such that the corresponding toric \nscheme ${\\mathbb P}^{\\prime}$ (viewed over ${\\mathcal O}[\\pi^{\\frac{1}{d}}]$)\nhas smooth general fiber, and special fiber a divisor with simple normal crossings.\n\nThen $(X,{\\mathbb P}^{\\prime})$ is also tropical, and the multiplication map\n$$m: {\\mathcal T} \\times_{{\\mathcal O}} {\\mathcal X}^{\\prime} \\rightarrow {\\mathbb P}^{\\prime}$$\nis smooth by the previous proposition.  Since the special fiber of ${\\mathbb P}^{\\prime}$ is a divisor\nwith simple normal crossings, so is the special fiber of ${\\mathcal T} \\times_{{\\mathcal O}} {\\mathcal X}^{\\prime}$.\nHence the special fiber of ${\\mathcal X}^{\\prime}$ is a divisor with simple normal crossings as well.\nSimilarly, the general fiber of ${\\mathcal X}^{\\prime}$ is smooth because the general fiber of\n${\\mathbb P}^{\\prime}$ is smooth.\n\\end{proof}\n\n\\begin{defn} We call a pair $(X,{\\mathbb P})$ of the sort produced by Proposition~\\ref{prop:degeneration}\na {\\em normal crossings pair}.  If $(X,{\\mathbb P})$ is a normal crossings pair, and $\\Sigma$\nis the polyhedral decomposition of $\\operatorname{Trop}(X)$ corresponding to ${\\mathbb P}$, we say that\n$\\Sigma$ is a {\\em normal crossings decomposition} of $\\operatorname{Trop}(X)$.\n\\end{defn}\n\n\\begin{rem} \\rm In much of what follows, we will often need to attach a normal\ncrossings pair to a sch\\\"on variety $X$ over ${\\mathcal O}$.  To do this we may need to replace\n${\\mathcal O}$ with a ramified extension ${\\mathcal O}[\\pi^{\\frac{1}{d}}]$; this is harmless and we often\ndo so without comment.\n\\end{rem}\n\n\\section{Parameterized Tropical Varieties} \\label{sec:param}\n\nIn this section, given a sch\\\"on subvariety $X$ of ${\\mathcal T}$, we construct a\nnatural parameterization of $\\operatorname{Trop}(X)$ by a topological space $\\Gamma_X$.  This\nparameterization is functorial in a sense we make precise below.  Moreover, we\nwill see in the next section that the space $\\Gamma_X$ encodes more precise information\nabout the cohomology of $X$ than $\\operatorname{Trop}(X)$ does.  Our approach generalizes a construction\nof Speyer (\\cite{Sp3}, proof of Theorem 10.8) when $X$ has dimension $1$.\n\nSuppose we have a normal crossings pair $(X,{\\mathbb P})$ so $\\operatorname{Supp}(\\Sigma)=\\operatorname{Trop}(X)$.  We associate to $(X,{\\mathbb P})$\na polyhedral complex $\\Gamma_{(X,{\\mathbb P})}$ as follows: its $k$-cells are pairs\n$(P,Y)$, where $P$ is a $k$-dimensional polyhedron in $\\Sigma$ and $Y$ is an irreducible component of\n${\\mathcal X}_P$.  The cells on the boundary of $(P,Y)$ are the cells of the form $(P_i,Y_i)$,\nwhere $P_i$ is a facet of $P$ and $Y_i$ is the irreducible\ncomponent of ${\\mathcal X}_{P_i}$ containing $Y$ (there is exactly one such irreducible component\nbecause ${\\mathcal X}_{P_i}$ is smooth, so its irreducible components do not meet).  The complex \n$\\Gamma_{(X,{\\mathbb P})}$ maps naturally to $\\Sigma$ by sending $(P,Y)$ to $P$.\n\n\\begin{prop} \\label{prop:independence}\nThe underlying topological space of $\\Gamma_{(X,{\\mathbb P})}$ depends only on $X$.\n\\end{prop}\n\\begin{proof}\nAny two polyhedral decompositions of $\\operatorname{Trop}(X)$ have a common refinement; we can further\nrefine this to be a normal crossings decomposition of $\\operatorname{Trop}(X)$.  It thus suffices to show that if \n$\\Sigma$ and $\\Sigma^{\\prime}$ are normal crossings decompositions of $\\operatorname{Trop}(X)$, with associated\nnormal crossings pairs $(X,{\\mathbb P})$ and $(X,{\\mathbb P}^{\\prime})$, and\n$\\Sigma^{\\prime}$ refines $\\Sigma$, then the underlying topological spaces of\n$\\Gamma_{(X,{\\mathbb P})}$ and $\\Gamma_{(X,{\\mathbb P}^{\\prime})}$ are homeomorphic.\n\nSince $\\Sigma^{\\prime}$ is a refinement of $\\Sigma$, we have a map ${\\mathbb P}^{\\prime} \\rightarrow {\\mathbb P}$.\nLet ${\\mathcal X}^{\\prime}$ be the degeneration corresponding to the pair $(X,{\\mathbb P}^{\\prime})$.\nIf $P$ is a polyhedron of $\\Sigma$, and $P^{\\prime}$ is a polyhedron of $\\Sigma^{\\prime}$\ncontained in $P$, then this map induces a map of ${\\mathcal X}^{\\prime}_{P^{\\prime}}$ to ${\\mathcal X}_P$.\nIn particular, for every pair $(P^{\\prime},Y^{\\prime})$ of $\\Gamma_{X,{\\mathbb P}^{\\prime}}$,\nthe image of $Y^{\\prime}$ in ${\\mathcal X}$ is contained in an unique irreducible component $Y$ of\n${\\mathcal X}_P$.  The map taking $(P^{\\prime},Y^{\\prime})$ to $(P,Y)$ is then a map of polyhedral\ncomplexes $\\Gamma_{(X,{\\mathbb P}^{\\prime})} \\rightarrow \\Gamma_{(X,{\\mathbb P})}.$\n\nWe have a commutative diagram:\n\\begin{diagram}\n\\Gamma_{(X,{\\mathbb P}^{\\prime})} & \\rTo & \\operatorname{Trop}(X) \\\\\n\\dTo & \\ruTo & \\\\\n\\Gamma_{(X,{\\mathbb P})} & & \n\\end{diagram}\nThe fiber of $\\pi: \\Gamma_{(X,{\\mathbb P})}\\rightarrow \\operatorname{Trop}(X)$ over a point $w$ is in canonical\nbijection with the set $C({\\mathcal X}_P)$ of (geometric) connected components of ${\\mathcal X}_P$.  Similarly, the fiber\nof $\\pi^{\\prime}: \\Gamma_{(X,{\\mathbb P}^{\\prime})} \\rightarrow \\operatorname{Trop}(X)$ over $w$ is in canonical bijection\nwith $C({\\mathcal X}^{\\prime}_{P^{\\prime}})$.  By Lemma~\\ref{lemma:comp}, both of these sets of connected components\nare in bijection with $C(\\operatorname{in}_w X)$; in fact, we have a commutative diagram:\n\\begin{diagram}\nC(\\operatorname{in}_w X) & \\rTo & (\\pi^{\\prime})^{-1}(w)\\\\ \n\\dEq & & \\dTo\\\\\nC(\\operatorname{in}_w X) & \\rTo & \\pi^{-1}(w)\n\\end{diagram}\nThus the map $\\Gamma_{(X,{\\mathbb P}^{\\prime})} \\rightarrow \\Gamma_{(X,{\\mathbb P})}$\nis bijective, and is therefore a homeomorphism on the underlying topological\nspaces of $\\Gamma_{(X,{\\mathbb P}^{\\prime})}$ and $\\Gamma_{(X,{\\mathbb P})}$.\n\\end{proof}\n\nIn light of this proposition, we denote by $\\Gamma_X$ the underlying topological\nspace of $\\Gamma_{(X,{\\mathbb P})}$ for {\\em any} normal crossings pair $(X,{\\mathbb P})$.  \nWe think of $\\Gamma_X$, together\nwith its natural map to $\\operatorname{Trop}(X)$, as a ``parameterized tropical variety.''\nNote that $\\Gamma_X$ inherits an integral affine structure by pullback from\n$\\operatorname{Trop}(X)$; more precisely, for any normal crossings pair $(X,{\\mathbb P})$,\nthe map $\\Gamma_X \\rightarrow \\operatorname{Trop}(X)$ is\nlinear on any polyhedron in $\\Gamma_{(X,{\\mathbb P})}$.\n\nNote that for any $w \\in \\operatorname{Trop}(X)$, the number of preimages\nof $w$ in $\\Gamma_X$ is equal to the number of connected components of\n$\\operatorname{in}_w X$.  Therefore, if $\\Sigma$ is a normal crossings decomposition \nof $\\operatorname{Trop}(X)$, and $w$ is in the relative interior\nof a top dimensional cell $P$ of $\\Sigma$, then the number of preimages of\n$w$ is equal to the {\\em multiplicity} of $P$ in $\\operatorname{Trop}(X)$.  This suggests\nthat we should give $\\Gamma_X$ the structure of a weighted polyhedral complex\nby giving every polyhedron on $\\Gamma_X$ weight one.  \n\nIf we do this, then $\\Gamma_X$ satisfies a ``balancing condition'' analogous\nto the well-known balancing condition on $\\operatorname{Trop}(X)$.  Fix a normal\ncrossings decomposition $\\Sigma$ of $\\operatorname{Trop}(X)$, with corresponding normal\ncrossings pair $(X,{\\mathbb P})$.  Consider a polyhedron $(P,Y)$ of $\\Gamma_{(X,{\\mathbb P})}$\nof dimension $\\dim X - 1$, and let $\\{(P_i,Y_i)\\}$ be the top dimensional\npolyhedra of $\\Gamma_{(X,{\\mathbb P})}$ containing $(P,Y)$.\n\nFix a point $w$ with rational coordinates in the relative interior of $P$, and let $V_P$\nbe the linear span, ${\\operatorname{Span}}(P - w)$.  Similarly, for each $P_i$,\nlet $V_i$ be the positive span of ${\\operatorname{Span}}^+(P_i - w)$.\nThen $V_i\/V_P$ is a ray in ${\\mathbb R}^n\/V_P$; this collection of rays is the fan attached to\nthe toric variety ${\\mathbb P}_P$.  Let $v_i$ be the smallest integer vector along the ray $V_i\/V_P$.\n\n\\begin{prop} \\label{prop:balancing}\nThe $v_i$'s satisfy the ``balancing property'':\n$$\\sum_{(P_i,Y_i)} v_i = 0.$$\n\\end{prop}\n\\begin{proof}\nTorus-equivariant rational functions on ${\\mathbb P}_P$ correspond to lattice vectors $u$ in\nthe space $({\\mathbb R}^n\/V_P)^*$ dual to ${\\mathbb R}^n\/V_P$.  The valuation of $u$ along the divisor\nof ${\\mathbb P}_P$ corresponding to $v_i$ is simply $u(v_i)$.\n\nNow restrict $u$ to the curve ${\\mathcal X}_P$.  For any polyhedron $P^{\\prime}$ of $\\Sigma$ containing\n$P$, ${\\mathcal X}_P$ intersects the boundary divisior ${\\mathbb P}_{P^{\\prime}}$ in one point for each\ncell $(P_i,Y_i)$ of $\\Gamma_{(X,{\\mathbb P})}$ with $P_i = P^{\\prime}$.\nThe divisor of $u$ is therefore equal to\n$$\\sum_{(P_i,Y_i)} u(v_i) Y_i,$$\nas ${\\mathcal X}_P$ intersects each boundary divisor ${\\mathbb P}_{P_i}$ transversely.\nThis divisor is a principal divisor and thus has degree zero.\n\\end{proof}\n\nWe have thus attached to any sch\\\"on subvariety $X$ of ${\\mathcal T}$, a canonical, multiplicity\nfree parameterization by the topological space $\\Gamma_X$.  Moreover, this construction is \nfunctorial:  let ${\\mathcal T}$ and ${\\mathcal T}^{\\prime}$ be tori over ${\\mathcal O}$, and let\n$T$ and $T^{\\prime}$ be their general fibers.  Suppose we have sch\\\"on subvarieties\n$X$ and $Y$ of $T$ and $T^{\\prime}$, respectively, and a homomorphism of tori\n$T \\rightarrow T^{\\prime}$ that takes $X$ to $Y$.  We then have a natural map\n$f:\\operatorname{Trop}(X) \\rightarrow \\operatorname{Trop}(Y)$.\n\n\\begin{prop} \\label{prop:functoriality}\nThere is a natural map\n$\\Gamma_X \\rightarrow \\Gamma_Y$ that makes the diagram\n\\begin{diagram}\n\\Gamma_X & \\rTo & \\Gamma_Y\\\\\n\\dTo & & \\dTo\\\\\n\\operatorname{Trop}(X) & \\rTo & \\operatorname{Trop}(Y)\n\\end{diagram}\ncommute.\n\\end{prop}\n\\begin{proof}\nLet $\\Sigma^{\\prime}$ be a normal crossings decomposisition of $\\operatorname{Trop}(Y)$. \nBy proposition~\\ref{prop:degeneration}\nwe can find a normal crossings decomposition $\\Sigma$ of $\\operatorname{Trop}(X)$ \nsuch that the image of any cell of $\\Sigma$ under the map $f$ \nis contained in a cell of $\\Sigma^{\\prime}$.\nLet $(X,{\\mathbb P})$ and $(Y,{\\mathbb P}^{\\prime})$ be the tropical pairs corresponding to\n$\\Sigma$ and $\\Sigma^{\\prime}$, and let ${\\mathcal X}$ and ${\\mathcal Y}$ denote the associated\ntropical degenerations.  Since each cell of $\\Sigma$ maps into a cell of\n$\\Sigma^{\\prime}$, we obtain a map from ${\\mathcal X}$ to ${\\mathcal Y}$ extending the map\n$X \\rightarrow Y$.\n\nNow let $P$ be a polyhedron in $\\Sigma$, and $P^{\\prime}$ be the polyhedron\nof $\\Sigma^{\\prime}$ containing the image of $P$.  Then our map\n${\\mathcal X} \\rightarrow {\\mathcal Y}$ induces a map ${\\mathcal X}_P \\rightarrow {\\mathcal Y}_{P^{\\prime}}$.\n\nIf $(P,X_i)$ is a polyhedron of $\\Gamma_{(X,{\\mathbb P})}$, then by definition\n$X_i$ is a connected component of ${\\mathcal X}_P$.  The image of $X_i$\nin ${\\mathcal Y}_{P^{\\prime}}$ is contained in a connected component $Y_i$ of\n${\\mathcal Y}_{P^{\\prime}}$.  We can thus construct a map of polyhedral complexes\n$$\\Gamma_{(X,{\\mathbb P})} \\rightarrow \\Gamma_{(Y,{\\mathbb P}^{\\prime})}$$\nthat maps $(P,X_i)$ to $(P^{\\prime},Y_i)$ by the map $P \\rightarrow P^{\\prime}$.\nThe induced map $\\Gamma_X \\rightarrow \\Gamma_Y$ on underlying topological\nspaces is clearly continuous and makes the diagram commute.  \n\nTo see that it is independent of choices, let $\\pi_X$ and $\\pi_Y$\nbe the projections of $\\Gamma_X$ and $\\Gamma_Y$ to $\\operatorname{Trop}(X)$ and $\\operatorname{Trop}(Y)$\nrespectively.  We then have canonical bijections between\n$\\pi_X^{-1}(w)$ and $C(\\operatorname{in}_w X)$, and between $\\pi_Y^{-1}(f(w))$ \nand $C(\\operatorname{in}_{f(w)} Y)$.  The map $X \\rightarrow Y$ induces a natural map\n$\\operatorname{in}_w X \\rightarrow \\operatorname{in}_{f(w)} Y$, and the diagram\n\\begin{diagram}\nC(\\operatorname{in}_w X) & \\rTo & \\pi_X^{-1}(w)\\\\\n\\dTo & & \\dTo\\\\\nC(\\operatorname{in}_{f(w)} Y) & \\rTo & \\pi_Y^{-1}(f(w))\n\\end{diagram}\ncommutes.  As the left hand side is independent of the choices of $\\Sigma$\nand $\\Sigma^{\\prime}$, the result follows.\n\\end{proof}\n\n\\begin{rem} \\rm Although Proposition~\\ref{prop:functoriality} is stated for\nmaps $X \\rightarrow Y$ that are {\\em monomial morphisms} (i.e., that arise from\nmorphisms of the ambient tori), we can avoid this issue if $X$ and $Y$\nare intrinsically embedded.  Recall that $X$ is {\\em very affine} if it can\nbe embedded as a closed subscheme of a torus $T$.  In this case (c.f.\n~\\cite{Tevelev}, section 3) there is an intrinsic torus $T_X$ associated to $X$\na {\\em canonical} embedding of $X$ in $T_X$.  Moreover, if $X$ and $Y$ are\nvery affine and $f: X \\rightarrow Y$ is a morphism, there is a morphism\nof tori $T_X \\rightarrow T_Y$ that induces $f$.\n\\end{rem}\n\nWe also record, for later use, the following result relating the cohomology of\n$\\Gamma_X$ to that of $\\operatorname{Trop}(X)$:\n\n\\begin{lemma} \\label{lemma:cohomology}\nLet $X$ be sch\\\"on, and let $\\Sigma$ be a normal crossings decomposition of $\\operatorname{Trop}(X)$.\nSuppose that for each polyhedron $P$ in $\\Sigma$, ${\\mathcal X}_P$ is either connected or\nhas dimension zero.  Then the natural\nmap \n$$H^r(\\operatorname{Trop}(X),{\\mathbb Z}) \\rightarrow H^r(\\Gamma_X,{\\mathbb Z})$$\nis an isomorphism for $0 \\leq r < \\dim X$, and an injection for $r = \\dim X$.\n\\end{lemma}\n\\begin{proof}\nLet $(X,{\\mathbb P})$ be the normal crossings pair attached to $\\Sigma$, so that\n$\\Gamma_{(X,{\\mathbb P})}$ is a triangulation of $\\Gamma$.  The polyhedra $P$ in $\\Gamma_{(X,{\\mathbb P})}$\nwith $\\dim P < \\dim X$ are, by construction, in bijection with the polyhedra in $\\Sigma$  \nwith $\\dim P < \\dim X$.  Thus $\\Gamma_{(X,{\\mathbb P})}$ is obtained from $\\Sigma$ by adding\nadditional top-dimensional cells; the result follows immediately.\n\\end{proof}\n\n\\section{Weight filtrations and the weight spectral sequence} \\label{sec:weight}\n\nOur goal will be to relate the combinatorial structure of $\\Gamma_X$ to geometric\ninvariants of $X$.  The invariants that appear arise from Deligne's theory of weights, which\nwe now summarize.  Recall (c.f.~\\cite{WeilII}, 1.2) that if $F$ is a finite field of order $q$, a continuous $l$-adic \nrepresentation $\\rho$ of $\\operatorname{Gal}(F^{\\operatorname{sep}}\/F)$ has weight $r$ if all the eigenvalues of the geometric \nFrobenius of $F$ are algebraic integers $\\alpha$, all of whose Galois conjugates have complex absolute \nvalue equal to $q^{r\/2}$.  If $A$ is a finitely generated ${\\mathbb Z}$-algebra, an {\\'e}tale sheaf \n${\\mathcal F}$ on $\\operatorname{Spec} A$ \nhas weight $r$ if for each closed point $s$ of $\\operatorname{Spec} A$, the stalk ${\\mathcal F}_s$ has weight $r$ when \nconsidered as a $\\operatorname{Gal}(k(s)^{\\operatorname{sep}}\/k(s))$-module.\n\nFollowing Ito (\\cite{Ito}, 2.2), we extend this definition to the case where $F$\nis a purely inseparable extension of a finitely generated extension of ${\\mathbb F}_p$ or ${\\mathbb Q}$.  For\nsuch $F$, one can find a finitely generated ${\\mathbb Z}$-subalgebra $A$ of $F$ such that $F$ is a purely\ninseparable extension of the field of fractions of $A$.  \n\nIn this setting, a representation $\\rho$ of $\\operatorname{Gal}(F^{\\operatorname{sep}}\/F)$ has weight $r$ if there is an \nopen subset $U$ of $\\operatorname{Spec} A$, and a smooth ${\\mathcal F}$ on $U$ of weight $r$, such that $\\rho$ arises from ${\\mathcal F}$\nby pullback to $\\operatorname{Spec} F$.  The Weil conjectures imply that for any proper smooth\nvariety $X$ over $F$, and any $l$ prime to the characteristic of $F$,\n$H^r_{\\mbox{\\rm \\tiny \\'et}}(X_{F^{\\operatorname{sep}}},{\\mathbb Q}_l)$ has weight $r$. \n\nWe henceforth assume that the residue field $k$ of ${\\mathcal O}$ is a purely inseparable\nextension of a finitely generated extension of ${\\mathbb F}_p$ or ${\\mathbb Q}$.  \nWe also fix an $l$ prime to the characteristic of $k$.\n\nLet $G$ be the absolute Galois group of the field $K$.  Then $G$ admits a\nsurjection $G \\rightarrow \\operatorname{Gal}(k^{\\operatorname{sep}}\/k)$, whose kernel is the inertia group $I_K$\nof $K$.  If $M$ is a $G$-module on which $I$ acts through a finite quotient,\nthere is a finite index subgroup $H$ in $G$ such that $H \\cap I$ acts trivially\non $M$.  Thus $\\operatorname{Gal}(k^{\\operatorname{sep}}\/k^{\\prime})$ acts on $M$ for some finite extension\n$k^{\\prime}$ of $k$.  We say $M$ is pure of weight $r$ if it has weight $r$ as a\n$\\operatorname{Gal}(k^{\\operatorname{sep}}\/k^{\\prime})$-module.  Note that this is independent of $k^{\\prime}$.\n\nThe {\\'e}tale cohomology of a variety over $K$ with semistable reduction has a filtration by subquotients\nwhich are pure in the above sense.  More precisely, let ${\\mathcal X}$ be a proper scheme over ${\\mathcal O}$, of relative \ndimension $n$,\nwhose fiber $X_K$ over $\\operatorname{Spec} K$ is smooth and whose fiber ${\\mathcal X}_k$ over $\\operatorname{Spec} k$ is a divisor with\nsimple normal crossings.  Then the Rapoport-Zink weight spectral sequence relates the {\\'e}tale cohomology \nof $X_{K^{\\operatorname{sep}}}$ to the geometry of the special fiber ${\\mathcal X}_k$.  More\nprecisely, let ${\\mathcal X}_{k^{\\operatorname{sep}}}^{(r)}$ denote the disjoint union of $(r+1)$-fold intersections\nof irreducible components of ${\\mathcal X}_{k^{\\operatorname{sep}}}$; it is smooth of dimension $n-r$ over $k^{\\operatorname{sep}}$.\nWe then have:\n\n\\begin{thm}[\\cite{RZ}, Satz 2.10; see also~\\cite{Ito}]\nThere is a spectral sequence:\n$$E_1^{-r,w+r} = \\bigoplus_{s \\geq \\max(0,-r)} H^{w-r-2s}_{\\mbox{\\rm \\tiny \\'et}}({\\mathcal X}_{k^{\\operatorname{sep}}}^{(2s + r)}, {\\mathbb Q}_l(-r-s))\n\\Rightarrow H^w_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}}, {\\mathbb Q}_l).$$\n\\end{thm}\n\nHere ${\\mathbb Q}_l(n)$ is the $n$th ``Tate twist'' of the constant sheaf ${\\mathbb Q}_l$; that is, it is the\ntensor product of ${\\mathbb Q}_l$ with the $n$th tensor power of the sheaf ${\\mathbb Z}_l(1)$, where\n${\\mathbb Z}_l(1)$ is the inverse limit of the sheaves $\\mu_{l^n}$ of $l$-power roots of unity.  Note that ${\\mathbb Z}_l(1)$\nis pure of weight $-2$.\n\nThe boundary maps of this spectral sequence are completely explicit, and can be described as follows: \nup to sign, they are direct sums of restriction maps\n$$H^i_{\\mbox{\\rm \\tiny \\'et}}(Y, {\\mathbb Q}_l(-m)) \\rightarrow H^i_{\\mbox{\\rm \\tiny \\'et}}(Y^{\\prime}, {\\mathbb Q}_l(-m))$$\nwhere $Y$ is an irreducible component of ${\\mathcal X}_{k^{\\operatorname{sep}}}^{(j)}$ and\n$Y^{\\prime}$ is an irreducible component of ${\\mathcal X}_{k^{\\operatorname{sep}}}^{(j+1)}$ contained in $Y$,\nor Gysin maps\n$$H^i_{\\mbox{\\rm \\tiny \\'et}}(Y^{\\prime}, {\\mathbb Q}_l(-m)) \\rightarrow H^{i+2}_{\\mbox{\\rm \\tiny \\'et}}(Y, {\\mathbb Q}_l(-m+1))$$\nwhere $Y$ and $Y^{\\prime}$ are as above.\n\nMore precisely, each term $E_1^{p,q}$ is a direct sum of terms of the form\n$H^i_{\\mbox{\\rm \\tiny \\'et}}(Y, {\\mathbb Q}_l(-m))$ for some irreducible component $Y$ of ${\\mathcal X}_{k^{\\operatorname{sep}}}^{(j)}$.\nIf $Y^{\\prime}$ is an irreducible component of ${\\mathcal X}_{k^{\\operatorname{sep}}}^{(j+1)}$, then we have:\n\\begin{itemize}\n\\item Whenever $H^i_{\\mbox{\\rm \\tiny \\'et}}(Y, {\\mathbb Q}_l(-m))$ is a direct summand of $E_1^{p,q}$, and\n$H^i_{\\mbox{\\rm \\tiny \\'et}}(Y^{\\prime}, {\\mathbb Q}_l(-m))$ is a direct summand of $E_1^{p+1,q}$, then\nthe corresponding direct summand of the boundary map $E_1^{p,q} \\rightarrow E_1^{p+1,q}$\nis (up to sign) the restriction\n$$H^i_{\\mbox{\\rm \\tiny \\'et}}(Y, {\\mathbb Q}_l(-m)) \\rightarrow H^i_{\\mbox{\\rm \\tiny \\'et}}(Y^{\\prime}, {\\mathbb Q}_l(-m)).$$\n\\item Whenever $H^i_{\\mbox{\\rm \\tiny \\'et}}(Y^{\\prime}, {\\mathbb Q}_l(-m))$ is a direct summand of $E_1^{p,q}$, and\n$H^i_{\\mbox{\\rm \\tiny \\'et}}(Y, {\\mathbb Q}_l(-m+1))$ is a direct summand of $E_1^{p+1,q}$, then\nthe corresponding direct summand of the boundary map $E_1^{p,q} \\rightarrow E_1^{p+1,q}$\nis (up to sign) the Gysin map\n$$H^i_{\\mbox{\\rm \\tiny \\'et}}(Y^{\\prime}, {\\mathbb Q}_l(-m)) \\rightarrow H^{i+2}_{\\mbox{\\rm \\tiny \\'et}}(Y, {\\mathbb Q}_l(-m+1)).$$\n\\end{itemize}\n\nWe refer the reader to example \\ref{ex:curves} for a description of the weight spectral\nsequence in the case when $X$ is a smooth curve.\n\nNote that the term $E_1^{-r,w+r}$ of the weight spectral sequence is pure of weight $w+r$.\nAs the only map between ${\\mathbb Q}_l$-sheaves that are pure of different weights is the zero map,\nthis implies that the weight spectral sequence degenerates at $E_2$.  Moreover, the\nsuccessive quotients of the filtration on $H^*_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}}, {\\mathbb Q}_l)$ induced\nby the weight spectral sequence are pure.  The filtration arising in this way is called\nthe {\\em weight filtration} on $H^*_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}}, {\\mathbb Q}_l)$.\n\nWe say a $G$-module $M$ is {\\em mixed} if $M$ admits an increasing $G$-stable filtration\n$$\\dots \\subset W_rM \\subset W_{r+1}M \\subset \\dots$$\nsuch that $W_rM\/W_{r-1}M$ has weight $r$ for all $r$.  (Such a filtration, if it exists, will\nbe unique.)  We say $M$ is mixed of weights between $r$ and $r^{\\prime}$ if $M$ is mixed\nand the quotients $W_iM\/W_{i-1}M$ are nonzero only when $r \\leq i \\leq r^{\\prime}$.\nThe above result shows that the cohomology of any scheme over $K$ with semistable reduction\nis mixed.  More generally, one has:\n\n\\begin{thm} (c.f.~\\cite{Ito}, 2.3) Let $X$ be a smooth, proper $n$-dimensional variety over \n$K$.  Then $H^r_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}}, {\\mathbb Q}_l)$ is mixed of weights between\n$\\max(0,2r-2n)$ and $\\min(2n,2r)$.\n\\end{thm}\n\\begin{proof}\nIf $X$ has strictly semistable reduction, i.e., $X$ is isomorphic to the general fiber of a \nscheme ${\\mathcal X}$ that is proper over ${\\mathcal O}$, and whose special fiber is a divisor with \nsimple normal crossings, then this follows from the weight spectral sequence.  \nThe general case follows by de Jong's theory of alterations~\\cite{dJ}.\n\\end{proof}\n\n\\begin{prop} \\label{prop:cohomology}\nLet $X$ be a smooth $n$-dimensional variety over $K$, and $\\overline{X}$ a compactification\nof $X$ such that $\\overline{X} - X$ is a divisor with simple normal crossings.  Then for $r \\leq n$,\n$H^r_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}}, {\\mathbb Q}_l)$ is mixed of weights between $0$ and $2r$, and the natural map\n$$W_0H^r_{\\mbox{\\rm \\tiny \\'et}}(\\overline{X}_{K^{\\operatorname{sep}}}, {\\mathbb Q}_l) \\rightarrow W_0H^r_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}}, {\\mathbb Q}_l)$$\nis an isomorphism.\n\\end{prop}\n\\begin{proof}\nLet $D$ be the divisor $\\overline{X} \\setminus X$, and let $\\overline{D}_1, \\dots, \\overline{D}_r$ be \nits irreducible components.  Let $X_i$ be the open subset $X \\setminus \n\\{\\overline{D}_1 \\cup \\dots \\cup \\overline{D}_i\\}$.  We proceed\nby induction on $i$; the case $i = 0$ is clear. \n\nSuppose the proposition is true for $i$.  Define \n$$D_i = \\overline{D}_{i+1} \\setminus \\{\\overline{D}_1 \\cup \\dots \\cup \\overline{D}_i\\},$$\nso that $X_i \\setminus X_{i+1} = D_i$.\nBy the inductive hypothesis the spaces $H^r_{\\mbox{\\rm \\tiny \\'et}}((X_i)_L, {\\mathbb Q}_l)$\nand $H^r_{\\mbox{\\rm \\tiny \\'et}}((D_i)_L, {\\mathbb Q}_l)$\nare mixed of weights between $0$ and $2r$ for $r \\leq n$.  We have a Gysin sequence:\n$$\n\\begin{array}{cccccc}\nH^{r-2}_{\\mbox{\\rm \\tiny \\'et}}((D_i)_L, {\\mathbb Q}_l(-1)) & \\rightarrow &\nH^r_{\\mbox{\\rm \\tiny \\'et}}((X_i)_L, {\\mathbb Q}_l) & \\rightarrow & \nH^r_{\\mbox{\\rm \\tiny \\'et}}((X_{i+1})_L, {\\mathbb Q}_l) & \\rightarrow \\\\ \nH^{r-1}_{\\mbox{\\rm \\tiny \\'et}}((D_i)_L, {\\mathbb Q}_l(-1)) & \\rightarrow & \\dots,\n\\end{array}\n$$\nand the first and last terms are mixed of weights between $2$ and $2r$.  It follows that\n$H^r_{\\mbox{\\rm \\tiny \\'et}}((X_{i+1})_L, {\\mathbb Q}_l)$ is mixed of weights between $0$ and $2r$ as\nrequired.  We also obtain an isomorphism\n$$W_0H^r_{\\mbox{\\rm \\tiny \\'et}}((X_i)_L, {\\mathbb Q}_l) \\cong W_0H^r_{\\mbox{\\rm \\tiny \\'et}}((X_{i+1})_L, {\\mathbb Q}_l)$$ \nand hence by induction the desired isomorphism\n$$W_0H^r_{\\mbox{\\rm \\tiny \\'et}}(\\overline{X}_L, {\\mathbb Q}_l) \\cong W_0H^r_{\\mbox{\\rm \\tiny \\'et}}(X_L, {\\mathbb Q}_l).$$\n\\end{proof}\n\n\\section{Cohomology of sch\\\"on varieties} \\label{sec:schon}\n\nThe control that tropical geometry gives over the degenerations of sch\\\"on subvarieties\n$X$ of $T$ has significant consequences on the level of cohomology.  In particular the theory\nof vanishing cycles allows one to relate the {\\'e}tale cohomology of a nice tropical compactification \nof $X$ to that of its tropical degeneration.  When the degeneration is a divisor with\nsimple normal crossings, this relationship is given by the Rapoport-Zink weight spectral\nsequence. \n\nWe apply this sequence in the setting of tropical geometry.  Let $X$ be sch\\\"on.  By\nProposition~\\ref{prop:degeneration} there is a polyhedral complex $\\Sigma$, with\nsupport equal to $\\operatorname{Trop}(X)$ and corresponding toric scheme ${\\mathbb P}$, such that the pair \n$(X,{\\mathbb P})$ is tropical, the corresponding compactification $\\overline{X}$ of $X$\nis smooth with simple normal crossings boundary, and the special fiber of the corresponding\ntropical degeneration ${\\mathcal X}$ is a divisor with simple normal crossings.\n\nThe polyhedral complex $\\Gamma_{(X,{\\mathbb P})}$ encodes the combinatorics of the special fiber\n${\\mathcal X}_{k^{\\operatorname{sep}}}$.  In particular ${\\mathcal X}_{k^{\\operatorname{sep}}}$ is a union of smooth \nconnected varieties ${\\mathcal X}_v$, where $v$ runs over the vertices of $\\Gamma_{(X,{\\mathbb P})}$.  The varieties\n${\\mathcal X}_{v_1}, \\dots, {\\mathcal X}_{v_r}$ meet if and only if $v_1, \\dots, v_r$ are the vertices of\na polyhedron in $\\Gamma_{(X,{\\mathbb P})}$.  \n[Note that since ${\\mathcal X}_k$ is a simple normal crossings divisor, if $Y_0, \\dots, Y_r$ intersect\nin codimension $r$ then they are the only irreducible components of ${\\mathcal X}_{k^{\\operatorname{sep}}}$ containing \ntheir intersection.]  \n\nWe have a natural map $\\Gamma_{(X,{\\mathbb P})} \\rightarrow \\Sigma$. \nSince $\\Sigma$ is a triangulation of $\\operatorname{Trop}(X)$, and $\\Gamma_{(X,{\\mathbb P})}$ is a triangulation\nof $\\Gamma_X$, this induces a natural map\n$$H^r(\\operatorname{Trop} X, {\\mathbb Q}_l) \\rightarrow H^r(\\Gamma_X, {\\mathbb Q}_l).$$\nBy the proof of Lemma \\ref{lemma:cohomology}, this map is an isomorphism if ${\\mathcal X}_P$ is connected for\nevery polyhedron $P$ in $\\Sigma$, or (equivalently) if $\\operatorname{in}_w X$ is connected\nfor every $w$ in $\\operatorname{Trop}(X)$. \n\n\\begin{thm} \\label{thm:main} There is a natural isomorphism \n$$H^r(\\Gamma_X, {\\mathbb Q}_l) \\cong W_0 H^r_{\\mbox{\\rm \\tiny \\'et}}(\\overline{X}_{K^{\\operatorname{sep}}},{\\mathbb Q}_l),$$\nand hence a natural map\n$$H^r(\\operatorname{Trop}(X), {\\mathbb Q}_l) \\rightarrow W_0 H^r_{\\mbox{\\rm \\tiny \\'et}}(\\overline{X}_{K^{\\operatorname{sep}}},{\\mathbb Q}_l).$$\nThis map is an isomorphism if ${\\mathcal X}_P$ is connected for\nevery polyhedron $P$ in $\\Sigma$.\n\\end{thm}\n\\begin{proof}\nThe bottom nonzero row of the $E_1$ term of the Rapoport-Zink spectral sequence (i.e., the $w=-r$ row)\nis the complex:\n$$\nH^0_{\\mbox{\\rm \\tiny \\'et}}({\\mathcal X}_{k^{\\operatorname{sep}}}^{(0)},{\\mathbb Q}_l) \\rightarrow\nH^0_{\\mbox{\\rm \\tiny \\'et}}({\\mathcal X}_{k^{\\operatorname{sep}}}^{(1)},{\\mathbb Q}_l) \\rightarrow\nH^0_{\\mbox{\\rm \\tiny \\'et}}({\\mathcal X}_{k^{\\operatorname{sep}}}^{(2)},{\\mathbb Q}_l) \\rightarrow \\dots\n$$\nin which the horizontal maps are restriction maps.\nThis is simply the coboundary complex of the polyhedral complex formed by the bounded \ncells of $\\Gamma_{(X,{\\mathbb P})}$.  This polyhedral complex is homotopy equivalent to \n$\\Gamma_{(X,{\\mathbb P})}$.  We thus have a natural isomorphism\n$$E_2^{r,0} \\cong H^r(\\Gamma_X, {\\mathbb Q}_l).$$\n\\end{proof}\n\n\\begin{rem} \\rm Theorem~\\ref{thm:main} shows in particular that the space\n$W_0 H^r_{\\mbox{\\rm \\tiny \\'et}}(\\overline{X}_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)$, which {\\em a priori} depends on\n$\\overline{X}$ and thus a choice of $\\Sigma$, in fact depends only on $X$ and is\n{\\em independent} of $\\Sigma$.  Proposition~\\ref{prop:cohomology} establishes this \ndirectly on the level of cohomology.\n\\end{rem}\n\nThe above results allow us to translate results about the cohomology of complete intersections \nin toric varieties into results about their tropicalizations.  For instance:\n\n\\begin{cor} \\label{cor:CI}\nLet $X$ be a sch\\\"on subvariety of $T$, and ${\\mathbb P}_K$ a smooth projective toric variety of $T$ such that:\n\\begin{enumerate}\n\\item the closure $Z$ of $X$ in ${\\mathbb P}_K$ is a smooth complete intersection of ample divisors, and\n\\item the boundary $Z \\setminus X$ is a divisor with simple normal crossings.\n\\end{enumerate}\nThen $H^r(\\Gamma_X, {\\mathbb Q}_l) = 0$ for $1 \\leq r < \\dim X$.\n\\end{cor}\n\\begin{proof}\nBy Proposition~\\ref{prop:degeneration} and Theorem~\\ref{thm:main} there is a tropical\npair $(X,{\\mathbb P}^{\\prime})$, with corresponding compactification $\\overline{X}$ of $X$, such that\n$H^r(\\Gamma_X, {\\mathbb Q}_l)$ is isomorphic to $W_0 H^r_{\\mbox{\\rm \\tiny \\'et}}(\\overline{X}_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)$.\nBy Proposition~\\ref{prop:cohomology} the latter is isomorphic to\n$W_0 H^r_{\\mbox{\\rm \\tiny \\'et}}(Z_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)$.  \n\nSince $Z$ is a complete intersection in ${\\mathbb P}_K$, the Lefschetz hyperplane theorem shows that\nfor $r < \\dim X$, the restriction map\n$$H^r_{\\mbox{\\rm \\tiny \\'et}}({\\mathbb P}_{K^{\\operatorname{sep}}},{\\mathbb Q}_l) \\rightarrow H^r_{\\mbox{\\rm \\tiny \\'et}}(Z_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)$$ \nis an isomorphism.  But ${\\mathbb P}_K$ is a smooth toric variety, and hence has good reduction.\nThe weight spectral sequence thus shows that $W_0 H^r_{\\mbox{\\rm \\tiny \\'et}}({\\mathbb P}_{K^{\\operatorname{sep}}},{\\mathbb Q}_l) = 0$\nfor $r > 0$.\n\\end{proof}\n\nUnder more restrictive hypotheses on $X$, we can turn the above result into a result about the\ncohomology of $\\operatorname{Trop}(X)$.  This will be the main goal of section~\\ref{sec:CI}.\n\n\\section{Monodromy} \\label{sec:monodromy}\n\nIn many situations, the weight filtration has an alternative interpretation in terms of monodromy.\nLet $X$ be a variety over $K$, and consider the base change $X_{K^{\\operatorname{sep}}}$ of $X$ to $K^{\\operatorname{sep}}$.\nThe group $\\operatorname{Gal}(K^{\\operatorname{sep}}\/K)$ admits a map to $\\operatorname{Gal}(k^{\\operatorname{sep}}\/k)$; the kernel is the\ninertia group $I_K$.  The group $I_K$ is profinite; if $l$ is prime to the characteristic of $k$\nthen the pro-$l$ part $I_K^{(l)}$ of $I_K$ is isomorphic to ${\\mathbb Z}_l(1)$.  (The Tate twist here refers\nto the fact that the quotient $\\operatorname{Gal}(k^{\\operatorname{sep}}\/k)$ acts on $I_K^{(l)}$ by conjugation in the\nsame way that it acts on the inverse limit of the roots of unity $\\mu_{l^n}$.\n\nThe group $\\operatorname{Gal}(K^{\\operatorname{sep}}\/K)$ acts on the {\\'e}tale cohomology\n$H^i_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)$ for any prime $l$.  This action is quasi-unipotent, i.e.\na subgroup of $H$ of $I_K$ of finite index acts unipotently on $H^i_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)$.\n(And thus the action of $H$ factors through $I_K^{(l)}$.)\nIn particular there is a nilpotent map \n$$N: H^i_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l) \\rightarrow H^i_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l(-1))$$\ncalled the {\\em monodromy operator} such that for all $\\sigma \\in H$, $\\sigma$ acts on\n$H^i_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)$ by $\\exp(t_l(\\sigma)N)$, where $t_l$ is the map\n$I_K \\rightarrow I_K^{(l)} \\cong {\\mathbb Z}_l(1).$\n\nNow, if $V$ is any finite dimensional vector space, with a nilpotent endomorphism $N$ such that $N^r = 0$, \nthen there is a unique increasing filtration $\\{V_i\\}$ on $V$ such that:\n\\begin{itemize}\n\\item $V_r = V$,\n\\item $V_{-r} = 0$,\n\\item $N$ maps $V_i$ to $V_{i-2}$, and\n\\item $N^i$ induces an isomorphism $V_i\/V_{i-1} \\rightarrow V_{-i}\/V_{-i-1}$.\n\\end{itemize}\n(see~\\cite{WeilII} I, 1.7.2 for details.)  We thus obtain a natural filtration, called the\nmonodromy filtration, on $H^i_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}}, {\\mathbb Q}_l)$.\n\n\\begin{rem} \\rm If $V$ consists of a single Jordan block of dimension $r$, one sees easily that\n$V_i\/V_{i-1}$ is one-dimensional for $i \\in \\{r-1, r-3, \\dots, -r+1\\}$, and zero otherwise.  \nMoreover, $V_{r-1-2k}$ is the image of $N^k$ for $0 \\leq k \\leq r-1$.\nIt is thus straightforward to read off the filtration coming from an arbitrary $V$ and $N$ from\na Jordan normal form for $N$.  The filtration is independent of choices, even though the Jordan\nnormal form of $N$ is not.\n\\end{rem}\n\nWhen $X$ has a semistable model, one can read the monodromy action on $X$ off\nof the weight spectral sequence $E^{p,q}$.  More precisely, there is a\nmonodromy operator\n$N: E_1^{p,q} \\rightarrow E_1^{p+2,q-2}(-1)$, which converges to the the monodromy operator $N$\non $H^i_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}}, {\\mathbb Q}_l)$.  It is easily described:\nif $H^i_{\\mbox{\\rm \\tiny \\'et}}(Y,{\\mathbb Q}_l(-m))$ occurs as a direct summand of $E_1^{p,q}$, and\n$H^i_{\\mbox{\\rm \\tiny \\'et}}(Y,{\\mathbb Q}_l(-m+1))$ occurs as a direct summand of $E_1^{p+2,q-2}$, then\nthe corresponding direct summand of $N$ is the identity \n$$H^i_{\\mbox{\\rm \\tiny \\'et}}(Y,{\\mathbb Q}_l(-m)) \\rightarrow H^i_{\\mbox{\\rm \\tiny \\'et}}(Y,{\\mathbb Q}_l(-m+1))(-1).$$\nAll other direct summands of $N$ are the zero map.\n\nThe following conjecture (the ``weight-monodromy conjecture'') relates the weight filtration to the \nmonodromy filtration in this situation:\n\n\\begin{conj} \\label{conj:w-m}\nThe top nonzero power of the monodromy operator:\n$$N^r: E_2^{-r,w+r} \\rightarrow E_2^{r,w-r}$$\nis an isomorphism for all $r,w$.  In particular the weight filtration on $H^i_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)$\n$E$ coincides (up to a shift in degree) with the monodromy filtration; that is,\n$$H^w_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{-r}\/H^w_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{-r-1} \\cong \nW_{w-r} H^w_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l).$$\n\\end{conj}\n\nThe weight-monodromy conjecture is well-known to hold for curves and surfaces.  If\n${\\mathcal O}$ is an equal characteristic discrete valuation ring, it is a difficult theorem of\nIto (\\cite{Ito}, Theorem 1.1).  It is open in general when ${\\mathcal O}$ has mixed characteristic.\n\nFor the remainder of this section we assume we are in a situation where Conjecture~\\ref{conj:w-m}\nholds.  The following result, due to Speyer (\\cite{Sp3}, Theorem 10.8) for curves, follows \nimmediately:\n\n\\begin{cor} \\label{cor:bound} \nLet $b_r(\\Gamma_X)$ and $b_r(X)$ denote the $r$th Betti numbers of $\\Gamma_X$ and $X$,\nrespectively.  Then we have:\n$$b_r(\\Gamma_X) \\leq \\frac{1}{r+1} b_r(X).$$\n\\end{cor}\n\\begin{proof}\nTheorem~\\ref{thm:main}, together with the weight-monodromy conjecture, shows that $b_r(\\Gamma_X)$ \nis the dimension of $H^r_{\\mbox{\\rm \\tiny \\'et}}(\\overline{X}_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{-r}$.  The dimension of this\npiece of the monodromy filtration counts the number of Jordan blocks of size $r+1$ \nin a Jordan normal form for $N$ acting on $H^r_{\\mbox{\\rm \\tiny \\'et}}(\\overline{X}_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)$.  \nIn particular the dimension of the latter is at least $r+1$ times the dimension of the \nformer.\n\\end{proof}\n\nSuppose $X_K$ is an $n$-dimensional variety.  There is a geometric interpretation of the action of the $n$th power of the monodromy operator on the middle-dimensional cohomology.\n\\begin{prop} \\label{prop:npower} The $d$th power of the monodromy map acting on middle cohomology,\n\\[N^d:H^d_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{d}\/H^d_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{d-1}\\rightarrow H^d_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{-d}(-d).\\]\nis the map\n\\[H_d(\\Gamma_{(X,{\\mathbb P})},{\\mathbb Q}_l)(-d)\\rightarrow H^d(\\Gamma_{(X,{\\mathbb P})},{\\mathbb Q}_l)(-d)\\]\ninduced from the ``volume pairing'' on the parameterizing complex $\\Gamma_{(X,{\\mathbb P})}$ which takes a pair of (integral) $d$-dimensional cycles to the (oriented) lattice volume of their intersection.  \n\\end{prop}\n\n\\begin{proof}\nThe term $H^d_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{d}\/H^d_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)_{d-1}$ is computed by the $(-d,d)$-entry in the Rapoport-Zink spectral sequence.  The $d$th row is:\n$$\nH^0_{\\mbox{\\rm \\tiny \\'et}}({\\mathcal X}_{k^{\\operatorname{sep}}}^{(d)},{\\mathbb Q}_l)(-d) \\rightarrow\nH^2_{\\mbox{\\rm \\tiny \\'et}}({\\mathcal X}_{k^{\\operatorname{sep}}}^{(d-1)},{\\mathbb Q}_l)(-d+1) \\rightarrow\\dots\\rightarrow\nH^{2d}_{\\mbox{\\rm \\tiny \\'et}}({\\mathcal X}_{k^{\\operatorname{sep}}}^{(0)},{\\mathbb Q}_l)$$\nwhere the horizontal map is the Gysin map of ${\\mathcal X}^{(k)}\\rightarrow {\\mathcal X}^{(k-1)}$.  Since each component of ${\\mathcal X}^{(k)}$ is an $(d-k)$-dimensional smooth variety, this is the chain complex formed by the bounded cells of $\\Gamma_{(X,{\\mathbb P})}$.  Consequently, $E_2^{-k,d}\\cong H_k(\\Gamma,{\\mathbb Q}_l)(-k)$.  Now, \n\\[N^d:E_2^{-d,d}\\cong H_d(\\Gamma,{\\mathbb Q}_l)(-d)\\rightarrow E_2^{0,0}(-d)\\cong H^d(\\Gamma_{(X,{\\mathbb P})},{\\mathbb Q}_l)(-d)\\] is induced from the identity map on $H^0_{\\mbox{\\rm \\tiny \\'et}}({\\mathcal X}_{k^{\\operatorname{sep}}}^{(d)},{\\mathbb Q}_l)$.  In the language of homology and cohomology of $\\Gamma_{(X,{\\mathbb P})}$, it comes from the \nmap $C_d(\\Gamma_{(X,{\\mathbb P})})\\rightarrow C^d(\\Gamma_{(X,{\\mathbb P})})$ taking a simplex $F$ to the cocycle $\\delta_F:C^d(\\Gamma_{(X,{\\mathbb P})})\\rightarrow{\\mathbb Z}$ that is the indicator function of $F$.  Consequently, if we view $N^d$ as  a bilinear pairing on $H_d(\\Gamma,{\\mathbb Q}_l)$, it is the volume pairing as every bounded top-dimensional cell of $\\Gamma_{(X,{\\mathbb P})}$ has volume $1$.\n\\end{proof}\n\n\\begin{ex} \\label{ex:curves}\n\\rm Suppose that $X_K$ is a curve of genus $g$.  Then ${\\mathcal X}_{k^{\\operatorname{sep}}}^{(0)}$ is the \nnormalization of ${\\mathcal X}_{k^{\\operatorname{sep}}}$; it is a disjoint union of smooth curves $C_i$ of genus \n$g_i$.  On the other hand, ${\\mathcal X}_{k^{\\operatorname{sep}}}^{(1)}$ is the set of singular points of \n${\\mathcal X}_{k^{\\operatorname{sep}}}$; each such point lies on exactly two of the $C_i$.  The corresponding\nweight spectral sequence is nonzero only for $-1 \\leq r \\leq 1$ and $0 \\leq w+r \\leq 2$; it \nlooks like:\n$$\n\\begin{array}{ccccc}\n\\bigoplus_{p \\in {\\mathcal X}_{k^{\\operatorname{sep}}}^{(1)}} {\\mathbb Q}_l(-1) & \\rightarrow & \\bigoplus_i H^2_{\\mbox{\\rm \\tiny \\'et}}(C_i, {\\mathbb Q}_l) & & 0\\\\\n0 & & \\bigoplus_i H^1_{\\mbox{\\rm \\tiny \\'et}}(C_i,{\\mathbb Q}_l) & & 0\\\\\n0 & & \\bigoplus_i H^0_{\\mbox{\\rm \\tiny \\'et}}(C_i,{\\mathbb Q}_l) & \\rightarrow & \\bigoplus_{p \\in {\\mathcal X}_{k^{\\operatorname{sep}}}^{(1)}} {\\mathbb Q}_l\n\\end{array}\n$$\nThe sequence clearly degenerates at $E_2$.  \nThe monodromy operator $N$ is nonzero only from $E_1^{-1,2}$ to $E_1^{1,0}(-1)$; \nit is simply the identity map on $$\\bigoplus_{p \\in {\\mathcal X}_{k^{\\operatorname{sep}}}^{(1)}} {\\mathbb Q}_l(-1).$$\nWe thus find that the middle quotient of\nthe monodromy filtration on $H^1_{\\mbox{\\rm \\tiny \\'et}}(X_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)$ is isomorphic to\nthe direct sum of $H^1_{\\mbox{\\rm \\tiny \\'et}}(C_i,{\\mathbb Q}_l)$, whereas the top and bottom quotients are isomorphic\nto $H_1(\\Gamma,{\\mathbb Q}_l)$, (resp. $H^1(\\Gamma,{\\mathbb Q}_l)$) where $\\Gamma$ is the dual graph of \n${\\mathcal X}_{k^{\\operatorname{sep}}}$.  As above, the map $N: H_1(\\Gamma,{\\mathbb Q}_l) \\rightarrow H^1(\\Gamma,{\\mathbb Q}_l)$\ncan be interpreted as the length pairing on $H_1(\\Gamma,{\\mathbb Q}_l)$. \\end{ex}\n\nThis example has a more classical interpretation.  If we let $J$ be the Jacobian of\n$\\overline{X}$, then the connected component of the identity in the\nspecial fiber of the N{\\'e}ron model of $J$ over ${\\mathcal O}$ is an extension of an abelian variety\nby a torus; let $\\chi$ be the character group of this torus.  Then $\\chi$\nis naturally isomorphic to $H_1(\\Gamma,{\\mathbb Z})$.  Moreover, one has a monodromy pairing\n$\\chi \\times \\chi \\rightarrow {\\mathbb Z}$ (see \\cite{SGA} for details.)  If one identifies\n$\\chi$ with $H_1(\\Gamma,{\\mathbb Z})$, the resulting pairing on $H_1(\\Gamma,{\\mathbb Z})$ is precisely\nthe length pairing.\n\nTo summarize:\n\\begin{prop} If $X$ is a sch\\\"on open subset of a smooth proper curve $\\overline{X}$ over $K$, then\nthe ``length pairing'' on $\\Gamma_X$ coincides with the monodromy pairing on\nthe character group $\\chi$ associated to the Jacobian of $\\overline{X}$.\n\\end{prop}\n\nThis has connections to Mikhalkin's construction of tropical Jacobians.  Given a\ntropical curve $\\Gamma$, which Mikhalkin interprets as a metric graph, the length\npairing on $\\Gamma$ induces a map $H_1(\\Gamma) \\rightarrow \\operatorname{Hom}(H_1(\\Gamma),{\\mathbb Z})$; Mikhalkin\ndefines the tropical Jacobian of $\\Gamma$ to be the torus $\\operatorname{Hom}(H_1(\\Gamma),{\\mathbb R})\/H_1(\\Gamma)$.\nThis torus has a natural integral affine structure induced from that on $\\operatorname{Hom}(H_1(\\Gamma),{\\mathbb R})$.\nSee~\\cite{MZ} for details.\n\nMikhalkin's definition is purely combinatorial but has a nice interpretation in\nterms of the uniformization of abelian varieties: if $J$ is the Jacobian of\n$\\overline{X}$ then there is a pairing $\\chi \\times \\chi \\rightarrow \\overline{K}^*$\nwhose valuation is the monodromy pairing.  This pairing gives an embedding of\n$\\chi$ as a lattice in the torus $\\operatorname{Hom}(\\chi, \\overline{K}^*)$; the quotient\n$\\operatorname{Hom}(\\chi,\\overline{K}^*)\/\\chi$ is a rigid space isomorphic to $J$.\nIf we ``tropicalize'' this space by taking valuations, we obtain the\nspace $\\operatorname{Hom}(\\chi,{\\mathbb R})\/\\chi$, where $\\chi$ embeds into $\\operatorname{Hom}(\\chi,{\\mathbb R})$ by\nthe monodromy pairing.  In particular the ``tropicalization'' of $J$ is\nthe tropical Jacobian of $\\Gamma_X$.\n\nThe upshot is that- provided we are careful about what we mean by tropicalization-\n``tropicalization'' commutes with taking Jacobians.\n\nThe following result of \\cite{KMM} is another easy consequence of this point of view:\n\\begin{prop}[\\cite{KMM}, Theorem 6.4] Let $X$ be a sch\\\"on open subset of an\nelliptic curve $\\overline{X}$ over $K$ with potentially multiplicative reduction.\nThen $H_1(\\Gamma_X,{\\mathbb Z})$ is isomorphic to ${\\mathbb Z}$, and valuation of the $j$-invariant \n$j(\\overline{X})$ is equal to $-a$, where $a$ is the length of the unique cycle in $\\Gamma_X$.\n\\end{prop}\n\\begin{proof}\nReplacing ${\\mathcal O}$ with a suitable ramified extension we may assume that $\\Gamma_X$ is integral.\nThen $\\overline{X}$ has split multiplicative reduction.  This base change scales both $\\Gamma_X$ and\nthe valuation of $j(\\overline{X})$ by the degree $d$ of the extension.  Now the tropical\ndegeneration ${\\mathcal X}$ of $X$ associated to $\\Gamma_X$ gives a model of $\\overline{X}$ whose special\nfiber contains a cycle of rational curves, of length equal to the lattice length $a$ of the unique\ncycle in $\\Gamma_X$.  The conductor-discriminant formula (\\cite{Si}, Theorem 11.1) then shows that\nthe valuation of $j(X)$ is equal to $-a$.\n\\end{proof}\n\nIn fact, it is easy to see that any smooth curve $\\overline{X}$ contains a sch\\\"on open subset:\ntake a semistable model of $\\overline{X}$, embed it in ${\\mathbb P}^n_{{\\mathcal O}}$, let ${\\mathcal T}$ be the complement of\n$n+1$ hyperplanes in general positionin ${\\mathbb P}^n_{{\\mathcal O}}$, and take $X = {\\mathcal T} \\cap \\overline{X}$.  \nThen the compactification $\\overline{X}$ of $X$ in ${\\mathbb P}^n_{{\\mathcal O}}$ is tropical, and one verifies\neasily that the multiplication map is smooth.  Thus the above result applies to all elliptic curves\nwith potentially multiplicative reduction.\nLuxton and Qu~\\cite{LQ} have shown that any variety over a field of characteristic $0$ contains a\nsch\\\"on open subset.\n\n\\section{Complete Intersections} \\label{sec:CI}\n\nIn the constant coefficient case, a (Zariski) general hyperplane section of a sch\\\"on variety is\nsch\\\"on.  Unfortunately this is no longer true in the nonconstant coefficient case.  For instance,\nlet $X_k$ be a singular hypersurface in $T_k$.  Then any hypersurface $X$ in $T_K$ that reduces\nmodulo $\\pi$ to $X_k$ has $\\operatorname{in}_{(0,\\dots,0)} X = X_k$, and hence cannot be sch\\\"on.  The set\nof such $X$ is a rigid analytic open subset of the projective space of hypersurfaces of fixed degree.\n\nAs this example suggests, to study loci of sch\\\"on varieties in a nonconstant coefficient setting,\none needs to work with the rigid analytic topology\nrather than the Zariski topology.  (For the basics of the theory of rigid analytic spaces we refer the\nreader to~\\cite{EKL} or~\\cite{Schneider}; we use very little here.)  \n\nTo make precise the connection to rigid geometry, we first observe:\n\n\\begin{lemma} \\label{lemma:transversality}\nLet ${\\mathbb P}$ be a toric scheme, proper over ${\\mathcal O}$, and let $X$ be a subvariety of \nthe open torus $T$ in ${\\mathbb P} \\times_{{\\mathcal O}} \\operatorname{Spec} K$.\nSuppose that for all polyhedra $P$ in the polyhedral complex $\\Sigma$ corresponding to $P$, the \nclosure ${\\mathcal X}$ of $X$ in ${\\mathbb P}$ intersects ${\\mathbb P}_P$ transversely.  Then $X$ is sch\\\"on, and\n$(X, {\\mathbb P}^{\\prime})$ is a normal crossings pair, where ${\\mathbb P}^{\\prime}$ is the open subset\nof ${\\mathbb P}$ obtained by deleting all torus orbits that do not meet ${\\mathcal X}$.\n\nConversely, if $X$ is sch\\\"on and there exists a toric open subset ${\\mathbb P}^{\\prime}$ of ${\\mathbb P}$\nsuch that $(X,{\\mathbb P}^{\\prime})$ is a normal crossings pair, then the closure of $X$ intersects ${\\mathbb P}_P$ \ntransversely for all polyhedra $P$ in $\\Sigma$.\n\\end{lemma}\n\n\\begin{proof}\nConsider the multiplication map\n$$m:{\\mathcal T} \\times {\\mathcal X} \\rightarrow {\\mathbb P}^{\\prime}.$$\nIf $y$ is a point in ${\\mathbb P}^{\\prime}$ in the torus orbit corresponding to a polyhedron $P$ in\nthe subcomplex $\\Sigma^{\\prime}$ of $\\Sigma$ corresponding to ${\\mathbb P}^{\\prime}$, then the fiber over \n$y$ is isomorphic to the product\n${\\mathcal X} \\cap {\\mathbb P}^{\\prime}_P$ with a torus.  By assumption,\nthis is smooth, so $m$ has smooth fibers.  The argument of~\\cite{Hacking}, Lemma 2.6\nthen shows that $m$ is smooth.  It follows that $X$ is sch\\\"on and $(X,{\\mathbb P}^{\\prime})$\nis a normal crossings pair.  The converse is clear.\n\\end{proof}\n\nNote that the lemma implies that $\\operatorname{Trop}(X)$ will be equal to the support of $\\Sigma^{\\prime}$ \nfor all such $X$.  One can therefore use this result to study the space of sch\\\"on subvarieties of \na toric variety over $K$ with a given tropicalization.  We will not pursue this here, beyond a few\nstraightforward observations.\n\nSuppose ${\\mathbb P}$ is projective.  Fix an $X$ as in the lemma, and let $\\operatorname{Hilb}({\\mathbb P})$ be the Hilbert\nscheme over ${\\mathcal O}$ parameterizing subschemes of ${\\mathbb P}$ with the same Hilbert polynomial as the closure\nof $X$.  Complex points of $\\operatorname{Hilb}({\\mathbb P})$ correspond to subschemes of the special fiber of ${\\mathbb P}$; those \nthat meet each ${\\mathbb P}_P$ transversely form an open subset $U_0$ of \n$\\operatorname{Hilb}({\\mathbb P}) \\times_{{\\mathcal O}} \\operatorname{Spec} k$.\n\nNow if $y$ is a point of $\\operatorname{Hilb}({\\mathbb P})(\\overline{K})$, then $y$ corresponds to a subscheme $X_y$ of the \ngeneral fiber of ${\\mathbb P}$ over a finite extension of $K$.  Then $X_y \\cap T$ will satisfy the hypotheses\nof Lemma~\\ref{lemma:transversality} if, and only if, $y$ specializes to a point $y_0$\non the special fiber of $\\operatorname{Hilb}({\\mathbb P})$ that lies in $U_0$.  The set of points\n$y$ that specialize to $U_0$ forms a ``neighborhood of $U_0$'' in the rigid analytic topology\non $\\operatorname{Hilb}({\\mathbb P}).$  More precisely, let $\\operatorname{Hilb}({\\mathbb P})^{\\mbox{\\rm \\tiny rig}}$ denote the rigid analytic space\nassociated to the general fiber of $\\operatorname{Hilb}({\\mathbb P})$; then $\\operatorname{Hilb}({\\mathbb P})^{\\mbox{\\rm \\tiny rig}}$ is equipped\nwith a ``reduction mod $\\pi$'' map\n$$\\operatorname{Hilb}({\\mathbb P})^{\\mbox{\\rm \\tiny rig}} \\rightarrow \\operatorname{Hilb}({\\mathbb P}) \\times_{{\\mathcal O}} \\operatorname{Spec} k.$$\nThe preimage of $U_0$ under this map is an admissible open subset $U^{\\mbox{\\rm \\tiny rig}}$ of\n$\\operatorname{Hilb}({\\mathbb P})^{\\mbox{\\rm \\tiny rig}}$, and those $y \\in \\operatorname{Hilb}({\\mathbb P})(\\overline{K})$ such that $X_y \\cap T$ satisfies the \nhypotheses of Lemma~\\ref{lemma:transversality} are precisely the $\\overline{K}$-points of $U^{\\mbox{\\rm \\tiny rig}}$.\n\nIf we restrict our attention to complete intersections, we can say more than this.\nIn particular fix a projective toric scheme ${\\mathbb P}$ over ${\\mathcal O}$, and ample line\nbundles $L_1, \\dots, L_s$ on ${\\mathbb P}$.  The space ${\\mathcal H}$ parameterizing tuples\n$(D_1, \\dots, D_s)$ such that for each $i$, $D_i$ is an effective divisor in the linear system\ncorresponding to $L_i$, and all the $D_i$'s intersect transversely, is an open subset\nof a product of projective spaces over ${\\mathcal O}$.  \n\nBy Bertini's theorem, the set of points in ${\\mathcal H}(k)$ that correspond to divisors\n$(D_1, \\dots, D_s)$ in ${\\mathbb P} \\times_{{\\mathcal O}} \\operatorname{Spec} k$ such that $D_1 \\cap \\dots \\cap D_s$ \nintersects each stratum ${\\mathbb P}_P$ of ${\\mathbb P}$ transversely is an open dense subset $U_0$ of\nthe special fiber of ${\\mathcal H}$.  The preimage of $U_0$ under the reduction map\n$${\\mathcal H}^{\\mbox{\\rm \\tiny rig}} \\rightarrow {\\mathcal H} \\times_{{\\mathcal O}} \\operatorname{Spec} k$$\nis a (necessarily nonempty) admissible open subset $U^{\\mbox{\\rm \\tiny rig}}$ of ${\\mathcal H}^{\\mbox{\\rm \\tiny rig}}$; the\npoints of $U^{\\mbox{\\rm \\tiny rig}}$ correspond precisely to those complete intersections\n$D_1 \\cap \\dots \\cap D_s$ whose intersection with $T$ satisfies the conditions of \nLemma~\\ref{lemma:transversality}.\n\nMoreover, if $(D_1, \\dots, D_s)$ is a $K$-point of $U^{\\mbox{\\rm \\tiny rig}}$, and $X$ is the corresponding\ncomplete intersection $D_1 \\cap \\dots \\cap D_s \\cap T$ in $T$, then for each polyhedron $P$ in\n$\\Sigma$, ${\\mathcal X}_P = D_1 \\cap \\dots \\cap D_s \\cap {\\mathbb P}_P$ is the intersection of ample divisors\nin the smooth toric variety ${\\mathbb P}_P$, and is therefore either zero-dimensional or connected.\n\nLemma~\\ref{lemma:cohomology} and Theorem~\\ref{thm:main} now have immediate implications\nfor the cohomology of $\\operatorname{Trop}(X)$:\n\n\\begin{thm} \\label{thm:CI}\nLet $(D_1, \\dots, D_s)$ be a $K$-point of $U^{\\mbox{\\rm \\tiny rig}}$, and set\n$$X = D_1 \\cap \\dots \\cap D_s \\cap T.$$\nThen $H^r(\\operatorname{Trop}(X),{\\mathbb Q}_l)$ vanishes for $1 \\leq r < \\dim X$, and the natural map:\n$$H^r(\\operatorname{Trop}(X),{\\mathbb Q}_l) \\rightarrow W_0 H^r_{\\mbox{\\rm \\tiny \\'et}}(\\overline{X}_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)$$\nis injective for $r = \\dim X$.\n\\end{thm}\n\\begin{proof}\nThe above discussion shows that $X$ is sch\\\"on and $(X,{\\mathbb P})$ is a normal crossings pair.\nWe thus apply Theorem~\\ref{thm:main} and Lemma~\\ref{lemma:cohomology} to see that the\nmap\n$$H^r(\\operatorname{Trop}(X),{\\mathbb Q}_l) \\rightarrow W_0 H^r_{\\mbox{\\rm \\tiny \\'et}}(\\overline{X}_{K^{\\operatorname{sep}}},{\\mathbb Q}_l)$$\nis an isomorphism for $0 \\leq r < \\dim X$ and injective for $r = \\dim X$.  On the other\nhand, $\\overline{X}$ is a complete intersection in the general fiber of the smooth toric\nvariety ${\\mathbb P} \\times_{{\\mathcal O}} \\operatorname{Spec} K$.  The result thus follows from Corollary~\\ref{cor:CI}.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{Introduction}\n\\label{sec:intro}\nOver the past several years many dedicated experiments\nhave been used to detect\nthe Sunyaev--Zel'dovich (SZ) effect \\citep{Sunyaev1970}\nfrom galaxy clusters at radio wavelengths [e.g.,\nBerkeley-Illinois-Maryland Association (BIMA) \\citep{Dawson2006};\nCombined Array for Research in Millimeter-wave Astronomy (CARMA) \\citep{Muchovej2012, Mantz2014};\nthe South Pole Telescope (SPT) \\citep{Reichardt2013};\nthe N\\'eel IRAM KIDs Array (NIKA) \\citep{Adam2014};\nthe Atacama Pathfinder EXperiment Sunyaev--Zel'dovich Instrument (APEX-SZ) \\citep{Dobbs2006, Bender2016};\nthe Arcminute Microkelvin Imager (AMI) \\citep{Zwart2008, Rumsey2016};\n{\\em Planck} Surveyor  \\citep{PlanckCollaboration2011};\nthe Atacama Cosmology Telescope (ACT) \\citep{Hasselfield2013};\nArray for Microwave Background Anisotropy (AMiBA) \\citep{Lin2016}].\nWithin the next few years, new observational facilities\nwill become operational and will search for galaxy clusters,\ncomplementing the galaxy cluster census across\nthe Universe\n[e.g.,\nNew IRAM KID Array 2 (NIKA2) on\nthe Institut de Radio Astronomie Millimetrique 30~m telescope \\citep{Calvo2016}].\n\n\n\nThe One Centimeter Receiver Array (OCRA)\n\\citep{Browne2000,Peel2011} is one of the\nexperiments capable of detecting the SZ effect at 30~GHz using\nbeam-switching radiometers installed on a 32-meter radio\ntelescope \\citep{Lancaster2007,Lancaster2011}.\nOCRA will be mostly sensitive to SZ clusters with virial size $> 3'$\nand hence to clusters at redshifts in the range $0.1 < z < 0.5$ and with\nmasses $M_{\\rm{vir}} > 3\\times 10^{14} M_{\\odot}\/h$ \\citep{Lew2015}.\nHowever, a single\nfrequency, beam-switching system may suffer from confusion with\nthe primordial cosmic microwave background (CMB) or suffer from\nsystematic error when observing extended sources.\n\nConfusion effects due to the CMB were investigated in detail by\n\\cite{Melin2006} for AMI, SPT and {\\em Planck} Surveyor.  It was\nfound that for single frequency instruments, such as AMI (a\n15~GHz interferometer), the photometric accuracy that contributes\nto the accuracy of the reconstructed comptonization parameter is\nstrongly limited due to primary CMB confusion.\n\n\nIn \\cite{Lew2015} the impact of CMB flux density confusion at 30\nGHz was investigated, in particular for the OCRA\/RT32 (32~m Radio\nTelescope in Toru\\'n, Poland) experiment.  It was found that the\n$1\\sigma$ thermal SZ (tSZ) flux density uncertainty due to CMB\nconfusion should be of the order of $10\\%$ for the range of\nclusters detectable with OCRA.  However, in that work, the impact\non the reconstructed comptonization parameter in the presence of\nthe CMB and radio sources was not calculated directly for the\ncase of dual-beam differential observations.\n\n\nThe $\\approx 3'$ separation of OCRA beams is very effective in\nCMB removal, but large correcting factors are required to\ncompensate for the missing SZ signal (after accounting for point\nsources) \\citep{Lancaster2007}.  Thus, there is a trade-off\nbetween compromising photometry by the primary CMB signal versus\nlosing flux due to the differential beam pattern.  In between\nthese extremes, there should exist an optimal separation of\ndifferential beams that would need to be defined by criteria that\naim to maximize CMB removal and minimize SZ flux density removal.\n\nIn this paper, we reconsider the issue of systematic effects on\nthe reconstructed comptonization parameter from single frequency,\nbeam-switched observations performed with a cm~wavelength\nradiometer.  We consider a particular instrumental setting for\nthe OCRA\/RT32 experiment and an extension to the standard\nobservation scheme that previously involved only the angular\nscales defined by the receiver feeds. The extension adds\nadditional beam pointings that map cluster peripheries, further\nfrom the central core than the initial pointings.\n\nThe {\\em kinetic} SZ (kSZ) may significantly modify the\nbrightness of the cluster peripheries that are integrated with\nthe reference beam. The significance of this effect depends on a\ncombination of the peculiar velocities of the intra-cluster\nmedium (ICM) and internal gas clumps, but at cm wavelengths, the\nkSZ only weakly modifies the central brightness.\n\nWith dual-beam observations, the reference beam background\ncoverage improves while integrating along arcs around the cluster\ncenter as the field of view (FOV) rotates.  However, due to the\nsmall angular size of the arcs, the chance of zeroing the average\nbackground may be low, depending on the alignment with the CMB\npattern. We investigate the significance of this effect depending\non observational strategy.\n\nFor experiments limited by the size of the focal plane array the\nintegration time required to generate a radio map and to probe\nthe outer regions of a galaxy cluster is significant and can make\nthe observation prohibitive.  Therefore, previously, the method\nof reconstructing comptonization parameters from OCRA\nobservations of cluster central regions required inclusion of\nX-ray luminosity data in order to find the best fitting\n$\\beta$-model for each cluster, and correction for the SZ power\nlost due to the close beam separation.  However, this approach\nrelies on the cluster model assumptions and makes the radio SZ\nmeasurements dependent on X-ray measurements of the cluster.\nAnother possible approach is to observe SZ clusters out to larger\nangular distances but retain averaging over a range of\nparallactic angles.  This is done at the cost of incurring extra\nnoise due to weak tSZ in cluster peripheries and stronger\nsystematic effects due to CMB.\n\n\nAn OCRA-SZ observational program is presently underway. In\nsupport of this and similar efforts, we also investigate the\npossibility of mitigating CMB confusion by using the available\n{\\em Planck} data.  Finally, we calculate the astrometric\npointing and tracking accuracy requirements needed to attain a\ngiven accuracy in flux density reconstruction.\n\n\nIn Section~\\ref{sec:strategy} we review the current observing\nstrategy and discuss its possible extensions.  In\nSection~\\ref{sec:sims} we briefly outline our numerical\nsimulation setting.  Section ~\\ref{sec:CMBsims} describes the\nconstruction of CMB templates from the currently available {\\em\nPlanck} data.  Section~\\ref{sec:sample} describes simulated\nsamples of galaxy clusters used for the flux-density\nanalyses. The main results are in Sec.~\\ref{sec:results}.  Final\nremarks and conclusions are in Sections~\\ref{sec:discussion} and\n~\\ref{sec:conclusions} respectively.\n\n\\section{Observational strategy}\n\\label{sec:strategy}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{figure_1.eps}\n\\includegraphics[width=0.49\\textwidth]{figure_2.eps}\n\\includegraphics[width=0.49\\textwidth]{figure_3.eps}\n\\includegraphics[width=0.49\\textwidth]{figure_4.eps}\n\\caption{ ({\\em Top-left}) Projected map of the the line-of-sight\n(LOS) integrated comptonization parameter for a selected halo,\nand ({\\em top-right}) simulated CMB temperature fluctuations\n$\\Delta T$ including both primary CMB as well as tSZ and kSZ\neffects induced by the halo.  ({\\em Bottom-left}) 217~GHz {\\em\nPlanck} resolution simulation -- the Gaussian CMB $\\Delta T$\nsignal smoothed with a half-power beam width (HPBW) of $\\approx\n5'$ and including a realistic {\\em Planck} noise realization\n(see Sec.~\\ref{sec:CMBsims}). This map is used to make a high\nresolution CMB template by means of a smooth-particle\ninterpolation.  ({\\em Bottom-right}) Residual map: primary CMB\nincluding tSZ minus interpolated {\\em Planck} simulation\n(Sec.~\\ref{sec:CMBsims}).  The variance of the residual map\n(without SZ effects) is about one order of magnitude smaller\nthan that measured using maps that include CMB.  In the {\\em\ntop-right} panel the circles denote OCRA beamwidths\ntraversing over the background CMB for a typical observational\nscheme (see Sec.~\\ref{sec:strategy} for details).  In the {\\em\ntop-left} panel black circles represent the full focal plane\nof OCRA receivers.  }\n\\label{fig:templates}\n\\end{figure*}\n\n\nThe common position-switching mode of observing\n\\citep{Lancaster2011} is that in which the reference beam\nnon-uniformly (due to varying FOV rotation speed) integrates the\nbackground along arcs $\\approx 3'$ from the source.  In\nFig.~\\ref{fig:templates} ({\\em top-right} panel), circles denote\nOCRA beamwidths for a typical observational scheme\n\\citep{Birkinshaw2005}.  First, the beam pair ``A--B'' measures\nthe difference signal between the cluster center and periphery,\nrespectively.  The beam pair is then ``switched'',\ni.e. translated to configuration ``C--A'' with a swap of the\nroles of the primary and reference beams, so that beams ``C'' and\n``A'' now trace the cluster periphery and center, respectively.\nThe position switching cycle is closed by returning to the\ninitial configuration ``A--B'' and the cycle is repeated.\n\nAs the Earth rotates, the reference beams sweep arcs around the\ncluster center and probe different off-center background regions\n(beam B becomes B' and C becomes C').  The beam\nposition-switching reduces fluctuations due to atmospheric\nturbulence on time scales of a few tens of seconds.  At shorter\ntime scales fluctuations due to receiver-gain instability and\natmospheric absorption are reduced by switching and\ndifferentiating signals in receiver arms by means of\nelectronically-controlled phase switches\n\\citep{Peel2010,Lancaster2011}.\\footnote{Beam switching is\nrealized at the rate of 277~Hz which improves the $1\/f$ knee of\nthe resulting difference signal power spectrum roughly by an\norder of magnitude; typically down to frequencies $0.1{\\rm\nHz}<f_{\\rm knee}<1{\\rm Hz}$.}\n\nAn extension to this pattern can be realized by adding an extra\nbeam pointing ``D''.  In this case an observation cycle would be\nextended so that a beam pair would observe differences between\n``A--B'', then ``B--D'' and followed by ``A--B'', again probing\nthe cluster peripheries for varying parallactic\nangles.  This scheme can be extended to both sides of the cluster\nand to larger distances from the cluster center.  In\nFig.~\\ref{fig:templates} ({\\em top-left} panel) the projected\nbeamwidths of the OCRA-f receiver focal plane are shown (black\ncircles) overlaid on a nearby galaxy cluster seen through the SZ\neffect. The extra beam pointing ``D'' for the whole array is\nshown with blue circles.  In the following sections we\ninvestigate the implications of such an extended observation\nscheme using numerical simulations.\n\n\n\n\n\n\\section{Simulations}\n\\label{sec:sims}\n\n\n\\subsection{LSS and SZ effect}\n\\label{sec:LSSsims}\n\nFor the main results in this work we use the simulation approach\ndescribed in \\cite{Lew2015}, with a few modifications.  In\nparticular, for the same field of view ($\\approx 5.2^\\circ$) we use\nan increased map resolution of $\\approx 0.9''$.  In observational\npractise, often only the central comptonization parameter value\nis quoted, so we include the kSZ signal calculated as a\ncontribution to the measured Compton $y$-parameter for the\nappropriate frequency.\n\nIn this analysis we neglect the large-scale foreground\ngalactic synchrotron, free--free and dust emissions.  We \nassume these to be smooth enough to be \nremoved in differential observations.\n\nWe assume that atmospheric effects and receiver noise are\nmitigated by sufficiently long integrations (see\nSect.~\\ref{sec:strategy} and \\cite{Birkinshaw2005}).\nSystematic errors in\nflux density estimation that we neglect include feed and elevation dependent beam\nresponse, feed and elevation dependent sidelobes, and elevation\ndependent antenna gain.  We defer treatment of these effects to a\nseparate analysis of end-to-end full OCRA focal plane\nsimulations.  Although the simulated maps include some of the\neffects of halo--halo LOS projections, the significance of the\nprojection effects on the $y$-parameter photometry are not the\nmain focus of the present study.  These\neffects should be small in this study, since\nwe only consider the most massive systems at the\nFOV generation stage.\n\n\nA cluster is not necessarily observed at the position that\nmaximises the SZ signal.  For an X-ray selected galaxy cluster,\nSZ observations can be centered at the maximum of the X-ray\nsignal, but this may have a small offset with respect to the\nmaximum of the SZ signal, although the difference should be\nrather small.  This will effectively translate into pointing\nerrors that we model in this work.\n\nThe OCRA beam separation is $s_{\\rm OCRA} = 0.0526\\pm\n0.0010^\\circ$ (Fig.~\\ref{fig:templates}), which we adopt in this\nwork.  Within the measurement accuracy this does not depend on\nsky direction.\n\n\\subsection{CMB templates}\n\\label{sec:CMBsims}\n\nThe {\\em Planck} mission has provided full sky maps of the CMB\ntemperature fluctuations at several frequencies, including\n217~GHz, where the tSZ signal is minimal.  This opens up the\npossibility to correct the differential SZ observations at other\nfrequencies by removing a customized CMB template and thus reduce\nthe CMB confusion in single frequency observations, provided\nthat: (i) the small scale noise level of the template is low,\n(ii) the angular resolution of the data is sufficient and (iii)\ndiffuse Galactic foregrounds can be neglected.  In practise, \nrequirement (iii) may exclude all or most of the sky regions\ncovered by {\\em Planck} frequency maps on either side of the\nGalactic Plane.\n\nIn order to test the usefulness of {\\em Planck} data in\nminimizing the CMB-induced variance in the measured flux\ndensities with a single-frequency instrument, we generate CMB\nsimulations for {\\em Planck}'s 217~GHz frequency band using\nsimulated CMB power spectrum \\citep{Lewis2000} and assuming\ncosmological parameters as in \\cite{Lew2015}.  We assume a\nGaussian beam transfer function defined by a full-width half\nmaximum FWHM~=~$5.02'$ \\citep{PlanckCollaboration2016b} and use\nrealistic and publicly available 217~GHz {\\em Planck} receiver\nnoise\nsimulations\\footnote{\\protect{\\url{http:\/\/pla.esac.esa.int\/pla\/\\#maps}}}\nat Healpix \\citep{Gorski2005} resolution $n_s=2048$.\n\nWe simulate the primary CMB up to $\\ell_{\\rm\nmax}=3500$,\\footnote{The variance lost due to neglecting even\nhigher multipoles is negligibly small ($\\lesssim 0.004\\%$) and\nwell below the cosmic variance uncertainty.}  i.e. the OCRA\nbeam separation.  In order to create the final CMB template with\n$\\approx 0.9''$ resolution in small FOVs, we use a\nsmooth-particle interpolation of the projected field (as\ndiscussed in \\citealt{Lew2015}).\n\nReal observations at 217~GHz will contain kSZ contributions,\nwhich we ignore for generation of simulated templates.  For each\nrealization of the simulated template, we store maps with CMB\nsignal, and maps with the CMB smoothed with the instrumental beam\nand contributed by a {\\em Planck} noise realization\n(Fig.~\\ref{fig:templates}).  The former is used for simulating\nastrophysical signals (SZ) in FOVs, while the latter is used for\nremoving a {\\em Planck}-compatible version of CMB contamination,\nfor the same FOVs.  Each FOV simulation is made at fixed galactic\nlatitude $b=40^\\circ$, but at a different galactic longitudes to\naccount for variations due to direction dependent properties of\n{\\em Planck} noise.\n\nThe single frequency maps from {\\em Planck} mission are\ncontaminated by foregrounds other than the cluster tSZ and kSZ\nsignals, especially at low galactic latitudes.  Since in the\ncurrent work we do not investigate galactic foregrounds we\nadditionally analyze another set of simulations based on the\nforeground-reduced map generated with the needlets-based internal\nlinear combination (NILC) algorithm \\citep{PlanckCollaboration2016a}.  The\nalgorithm \\citep{Delabrouille2009} provides a very clean primary\nCMB map outside of masked regions which mask Galactic Plane and\nbright point sources (together $\\approx 3.6\\%$ of the full sky).\nWe simulate the NILC map using the published beam transfer\nfunction, and use the full mission NILC rendition of ringhalf-1\nand ringhalf-2 half difference maps to generate a {\\em Planck}\nNILC-compatible noise realization.  We verified that the\nresulting NILC simulations are compatible with the {\\em Planck}\nNILC map in terms of their angular pseudo-power spectra.  Slight\ndifferences in the high-$\\ell$ regime of up to a few percent are\npresent due to our choice of cosmological parameter values that\nis consistent with the WMAP9 results.\n\nThe NILC map resolution simulations are similar to those of the\n217~GHz map, so in Fig.~\\ref{fig:templates} we only show the map\nfor the 217~GHz case.\n\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{figure_5.eps}\n\\caption{ Distribution of the solid-angle--integrated\ncomptonization parameter $Y$ integrated within the angular\nradius $\\theta<0.75'$ from the cluster center ({\\em left});\ndistribution of cluster total mass measured within a volume\nwith mean mass density $500$ times higher than the critical\ndensity of the Universe at the cluster's redshift ({\\em\nmiddle}); and distribution of redshifts ({\\em right}) for the\nsimulated cluster samples (solid lines) and for the SPT cluster\nsample \\citep{Bleem2015} (dashed lines).  For each simulated\nsample only halos with $M_{\\rm vir}>M_{\\rm vir,\\min}$ and\n$z>z_{\\min}$ are chosen (see Table~\\ref{tab:samples}).  The\nvertical dashed line shows the division into high-$z$ and\nlow-$z$ sub-samples that is used later in the analysis.  }\n\\label{fig:sample}\n\\end{figure}\n\n\n\n\\subsection{Simulated galaxy cluster samples}\n\\label{sec:sample}\n\n\n\n\\begin{table}[t]\n\\caption{Selection criteria used for constructing galaxy cluster samples.\n}\n\\begin{ruledtabular}\n\\begin{tabular}{lcccc}\n\nParameter & \\multicolumn{2}{c}{Sample\/Value}\\\\\n& \\sampleHS & \\sampleDF \\\\\n$z_{\\min}$ & 0.05 & 0.0 \\\\\n$M_{\\rm vir,\\min}\\,[10^{14} M_\\odot\/h]$ & 4.0 & 2.0 \\\\\nhalo selection & full simulation volume& $5.2^\\circ\\times 5.2^\\circ$ FOV \\\\\nhalo count\\footnotemark[1] & 475 & 361 \\\\\nSub-samples &&&& \\\\\n\\multicolumn{1}{r}{low-$z\\leq0.4$} & 426 & 214  \\\\\n\\multicolumn{1}{r}{high-$z>0.4$}   & 49  & 147\n\n\\footnotetext[1]{~The actual number of halos\nused in statistical analyses are slightly different\nas they are further screened for halos that lie well within the\nprojected FOV, which is required for simulating dual-beam\nobservations at all possible parallactic angles and beam separations in a consistent way.}\n\\end{tabular}\n\\end{ruledtabular}\n\\label{tab:samples}\n\\end{table}\n\n\nFor the analyses presented in Sect.~\\ref{sec:results}, we\nconstruct two galaxy cluster samples. The first one, hereafter\nreferred as \\sampleHS (Fig.~\\ref{fig:sample} thick solid lines)\nis constructed by selecting the heaviest halos [$M_{\\rm vir}>\n4\\times 10^{14} M_\\odot\/h$ (see Table~\\ref{tab:samples})], from\neach independent simulation volume and using each recorded\nsimulation snapshot.  We impose a low redshift cut-off\n$z>z_{\\min}$ to remove very extended clusters.  The choice of\nredshifts for which simulation snapshots are taken is made such\nthat the simulation volume continuously fills comoving space out\nto the maximal redshift (see Fig.1 of~\\cite{Lew2015}).  For each\nsimulation volume we apply random periodic coordinate shifts of\nthe particles within, and we apply random coordinate switches.\nThis (i) improves redshift space coverage and (ii) yields cluster\nSZ surface brightness profiles in different projections, at the\ncost of generating a partially correlated sample.\n\nThe second sample, hereafter referred to as \\sampleDF\n(Fig.~\\ref{fig:sample} thin solid lines), is generated using a\nblind survey approach (as in ~\\cite{Lew2015}). We generate 37 FOV\nrealizations each $\\approx 27\\,{\\rm deg}^2$ together covering a sky\narea of $\\approx 1000\\, {\\rm deg}^2$.  From each realization we\nselect halos with virial masses $M_{\\rm vir,c} > 2\\times 10^{14}\nM_\\odot\/h$.\n\nThe solid angle integrated comptonization for any given halo\n(${\\protect Y=\\int y(\\hat {\\mathbf n}) d\\Omega}$) depends on a\ncombination of halo redshift and mass.  The \\sampleDF sample is\ndominated by lighter halos than those found in the SPT sample\n(Fig.~\\ref{fig:sample}), although redshift space distributions of\nthe two are similar. Hence, the bulk of the \\sampleDF sample\nhalos yields lower $Y(\\theta<0.75')$ values than those in the SPT\nsample (Fig.~\\ref{fig:sample}).  Although increasing the lowest\nmass limit for the halos of the \\sampleDF sample tends to make\nits mass and redshift distributions more consistent with those of\nthe SPT sample, it reduces the numbers of halos, thus increasing\nPoisson noise.  For the statistical analysis in\nthis work, larger simulations and more FOV realizations than are\ncurrently available would be required to reach\nconsistency. Therefore, we use this sample for tSZ analyses of\nsimulated dual beam observations, bearing in mind that in this\nlimit of weak SZ effects, CMB confusion is expected to be the\nmost significant. On the other hand, the \\sampleHS sample is\nexpected to be less affected by CMB confusion.\n\n\n\nIn order to investigate the differences between compact and\nextended SZ clusters we further split our cluster samples by\nredshift at $z=0.4$ (Table~\\ref{tab:samples}).  This split roughly\ncorresponds to half of the radial comoving distance to $z=1.0$,\nbeyond which we do not observe any heavy (Fig.~\\ref{fig:sample})\nhalos in our simulations.\nWe find that the low-$z$ and high-$z$\nsamples mainly differ due to the strength of the SZ effects,\nand due to the presence of sub-structures,\nbeing respectively stronger and more abundant in the low-$z$ subset.\nExamples for halos from low-$z$ and high-$z$ samples are shown\nbelow in Fig.~\\ref{fig:halos}.\n\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{figure_6.eps}\n\\caption{Simulated fractions ($F$) of SZ effect flux density at\n30~GHz recovered from difference, dual beam observations of\nclusters from \\sampleDF sample as a function of beam angular\nseparation $s$ and redshift range.  The shaded\/hatched regions\nmap the 68\\% confidence regions (CR) in the distribution of\n$F$.  The scatter in $F$ calculated from maps containing only\ntSZ signal is shown in gray.  The backslash-hatched region\nshows the effects of primary CMB on biasing the tSZ flux\ndensity measurements.  The forward-slash--hatched region shows\nthe intrinsic scatter due to kSZ when converted and embedded\ninto the 30~GHz thermal SZ effect maps.  The green region shows\nthe improvements in decreasing the intrinsic scatter in $F$ as\na result of subtracting the Planck CMB template from CMB+tSZ\nsimulated maps prior to flux density calculations.  The median\n$F$ values are shown as lines.  The 68\\% confidence regions\nabout the medians become asymmetric as the beam separation\nincreases (simulation sample error also becomes obvious in the\nTSZ+CMB case by comparing upper to lower plots; the TSZ,\nTSZ+KSZ, and TSZ+CMB cases are statistically equivalent between\nthe upper and lower panels).  The vertical dashed line marks\nthe actual separation of OCRA beams fixed by the telescope\noptics.  It is assumed that the reference beam covers an\nannulus around a galaxy cluster within parallactic angle range\n$[0^\\circ,180^\\circ]$ on either side of the central direction,\nand that pointing error $\\epsilon_p=0$ (see\nSec.~\\ref{sec:pointing}).  }\n\\label{fig:medianfDF}\n\\end{figure*}\n\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{figure_7.eps}\n\\caption{As in Fig.~\\ref{fig:medianfDF} but for the \\sampleHS sample.}\n\\label{fig:medianfHS}\n\\end{figure*}\n\n\n\\section{Analysis and results}\n\\label{sec:results}\n\n\n\n\\subsection{Systematic effects from beam separation}\n\\label{sec:beam_separation}\n\n\nDual beam difference observations capture only a fraction of the\nintrinsic flux density, depending on the physical extent of the\nsource, its redshift and the angular separation of the beams.  We\ndefine this fraction as\n\\begin{equation}\n\\label{eq:F}\nF(\\theta_b, s,q_{\\rm max}) = \\bigg\\langle \\frac{S_0^{\\rm x}(\\theta_b,s) - S_r^{\\rm x}(\\theta_b,s,\\mathbf{\\hat n_i})}{S_0^{\\rm tSZ}(\\theta_b)} \\bigg\\rangle_i,\n\\end{equation}\nwhere $S_0^{\\rm x}$ is the measured central flux density per beam\ninduced by effect ``x'', e.g. x~=~tSZ, $S_0^{\\rm tSZ}$ is the\ntrue central flux density per beam due to tSZ (neglecting CMB,\npoint sources, and other effects), $S_{\\mathrm r}^{\\rm x}$ is the\nflux density per beam due to effect ``x'' in the reference beam\ndirection ($\\mathbf{\\hat n}_i$), $\\theta_b$ is the instrumental\nhalf-power beam width (HPBW), and $s$ is the angular separation\nof the beams.  $N_{\\mathrm r}=500$ reference beam directions\n($\\mathbf{\\hat n}_i$) are chosen randomly from a uniform\ndistribution of parallactic angles ($q \\in [0,q_{\\max}]$), where\nthe upper limit $q_{\\max}$ is a free parameter.\n\nFor each halo, we measure these fractions $F$ by integrating\nspecific intensity directly from high resolution maps, and using\nthe mean over the $N_{\\mathrm r}$ values of $q$.  If there is no\nCMB contamination, i.e. setting x~=~tSZ, so that\n$S_0^{\\rm x} = S_0^{\\rm tSZ}$\nand\n$S_{\\mathrm r}^{\\rm x} = S_{\\mathrm r}^{\\rm tSZ}$,\nthen $F\\leq 1$ and $1-F$ represents the fraction of the signal\nlost only due to the closeness of the beam angular separation.\n\nThe impact of beam separation on the dual-beam observations is\nshown in Figs.~\\ref{fig:medianfDF} and~\\ref{fig:medianfHS} for\nthe \\sampleDF and \\sampleHS samples respectively, for an\nidealistic case of exact pointing---i.e., no pointing\ninaccuracies are allowed ($\\epsilon_p=0$).  In these figures, the\nmedian $F$ (from all halos matching the selection criteria) is\nplotted along with a 68\\% confidence region.  Clearly, dual-beam\nobservations at larger beam separations are less biased than\nobservations at smaller beam separations, and the 68\\% confidence\nrange generally shrinks as $s$ increases.\n\nThe significance of the primary CMB fluctuations for the\ndual-beam observations is estimated by setting x~= tSZ$+$CMB,\ni.e.,\n$S_0^{\\mathrm x}=S_0^{\\mathrm{tSZ+CMB}}$, \n$S_{\\mathrm r}^{\\mathrm x}=S_{\\mathrm r}^{\\mathrm{tSZ+CMB}}$.\nWhile the median $F$ does not differ significantly from the pure\ntSZ case, the 68\\% confidence region significantly increases with\nbeam separations due to primary CMB confusion.  For example,\nsince a primordial CMB fluctuation has a good chance of being of\nthe same sign as the SZ signal at the cluster center but of the\nopposite sign in a distant reference beam, $F$ can easily be\ngreater than unity, as is clear in Fig.~\\ref{fig:medianfDF}.  As\nexpected, the increase is stronger in the \\sampleDF\nsample\/high-$z$ sub-sample than in the \\sampleHS sample\/low-$z$\nsub-sample, due to differences in amplitudes of SZ effects\ncompared to the level of CMB fluctuations.\n\n\n\nComparing Figs.~\\ref{fig:medianfDF} and ~\\ref{fig:medianfHS} it\nis clear that the main difference is the relative significance of\nthe CMB as a source of confusion and the amount of residual\nbiasing.  However, for any individual high-$z$ and\/or low-mass\ncluster observation, the measured flux density can be biased\nsubstantially. \nThis can be inferred from the size of the\n$1\\sigma$ tSZ+CMB confidence region. Even observations of the\nmost massive clusters, which are the least affected by the\npresence of the CMB, can be biased substantially depending on the\nangular scales being measured ($s$) (Fig.~\\ref{fig:medianfHS}\nleft panels).  In the figure, the trade-off\nbetween CMB confusion due to observations at larger angular\nscales and the level of biasing ($F$) in the limit of small $s$\nis clearly seen.\n\n\nFor clusters that are small relative to the beam size,\nmeasurements far away from the cluster center are not really\nneeded as the $F$ values approach unity relatively fast\n(e.g. \\sampleDF\/ high-$z$ sample in Fig.~\\ref{fig:medianfDF}).\nAt the OCRA beam separation (the vertical line in the figures)\nthe primary CMB does not strongly contribute to the scatter in\nflux density measurements.  This is even more so in the case of\nthe \\sampleHS sample of heavy and low-$z$ clusters.  On the other\nhand, the most massive halos (Fig.~\\ref{fig:medianfHS}) require\nsignificant ($> 10\\%$) flux density corrections even at\nlarge beam separations (although these may partially be generated\nby projection effects discussed in Sec.~\\ref{sec:discussion}).\n\nIt is clear that in the two cluster samples, kSZ \nonly slightly increases the scatter in $F$ at the OCRA beam\nseparation, as expected at 30~GHz.\n\n\nThe impact of {\\em Planck} based CMB template removal is shown in green.\nThe calculation is done by setting x~= tSZ$+$CMB$-$template \nin Eq.~\\ref{eq:F},\ni.e.,\n$S_0^{\\mathrm x}=S_0^{\\mathrm{tSZ+CMB-template}}$, \n$S_{\\mathrm r}^{\\mathrm x}=S_{\\mathrm r}^{\\mathrm{tSZ+CMB-template}}$.\nFrom Figs.~\\ref{fig:medianfDF} and~\\ref{fig:medianfHS} it is\nclear that at the OCRA beam separation, and for the full range of\nparallactic angles, the {\\em Planck} template does not\nsignificantly help, or does not help at all, in reducing the\nconfusion due to primary CMB. However, in observations that probe\nlarger angular separations, the CMB template removal can\nsubstantially reduce the $1\\sigma$ contours. The template removal\nmay also be useful for observations of high-$z$ massive clusters\nfor which mapping larger angular distances away from the central\ndirections still appears to be well motivated. Both in the\nhigh-$z$ and low-$z$ sub-samples of the \\sampleHS sample the\ntemplate reduces the tSZ+CMB scatter nearly down to the level\nlimited by the intrinsic tSZ scatter for the full range of $s$\nstudied here (Fig.~\\ref{fig:medianfHS}).\n\n\n\n\\subsection{Parallactic angle dependence}\n\\label{sec:PA}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{figure_8.eps}\n\\includegraphics[width=0.49\\textwidth]{figure_9.eps}\n\\caption{As in Figs~\\protect\\ref{fig:medianfDF} and\n\\protect\\ref{fig:medianfHS}, but for observations where the\nreference beam covers an annulus around a galaxy cluster within\nparallactic angle range $[0^\\circ,22.5^\\circ]$ on either side\nof the cluster direction ({\\rm left}), or only on one side of\nthe cluster direction ({\\rm right}) for the \\sampleDF (top) and\n\\sampleHS (bottom) samples.  }\n\\label{fig:F_vs_PA_DF}\n\\end{figure}\nIn practise, the OCRA observations exploit beam and position\nswitching (Sect.~\\ref{sec:strategy}) but it is unrealistic to\ncover the full parallactic angle range: i.e. $q\\in\n[0^\\circ,q_{\\max}]$ where $q_{\\max}=360^\\circ$.\n\nIt was already known that position switching\n\\citep{Birkinshaw2005} significantly mitigates atmospheric\ninstabilities over the time scale of tens of seconds by (i)\nsubtracting linear drifts caused by large-scale precipitable\nwater vapor (PWV) fluctuations \\citep{Lew2016}, (ii) accounting\nfor beam response asymmetries, and (iii) maximizing the\nprobability of avoiding (masking out) intervening radio sources\nthat can significantly bias the SZ measurement.  In this section,\nwe show that position switching is also efficient in mitigating\nthe confusion due to primordial CMB, even with a very modest\ncoverage of parallactic angles.\n\nThere should not be any statistical correlation between\nprimordial CMB fluctuations and the locations of heavy\nhalos. Moreover, galaxy clusters have small angular sizes\ncompared those representing most of the CMB power.  Thus,\nclusters should mostly lie on slopes rather than peaks or troughs\nin the CMB map. Hence, sampling SZ flux density differences at\nopposite sides of a galaxy cluster core should help average out\nthe primordial CMB in comparison to one-sided observations.  We\nconfirm that this is indeed the case and find that this\nimprovement is reached at even moderate values of $q_{\\max}$.\n\nWe calculate $F(s)$ [Eq.~(\\ref{eq:F})] for maps containing tSZ\nand CMB using mean flux density estimates either according to the\nposition switching observation scheme or without it.  As before,\neach measurement is an average of $500$ dual-beam pointings at\ndifferent $q$ but drawn randomly from within the range\n$[0^\\circ,q_{\\max}]$ where\n$q_{\\max}\\in\\{180^\\circ,90^\\circ,45^\\circ,22.5^\\circ\\}$.\n\nThe result is shown in Fig.~\\ref{fig:F_vs_PA_DF} for\n$q_{\\max}=22.5^\\circ$.  By comparing the left panel of this\nfigure with the top-left panel of Fig.~\\ref{fig:medianfDF} it is\nclear that even strongly incomplete coverage of the parallactic\nangles does not cause significant broadening of the 68\\%\nconfidence level (CL) contours.  However, when position switching\nis not used (right panels in Fig.~\\ref{fig:F_vs_PA_DF}), the\nconfusion due to primordial CMB is stronger. As before, the\n\\sampleHS sample of the heaviest halos is less affected by the\npresence of the CMB, but the effect of not using position\nswitching is still visible, even at $s=s_{\\rm OCRA}$ (vertical\nline in Fig.~\\ref{fig:F_vs_PA_DF}, bottom panels).\n\n\n\\subsection{Systematic effects in redshift space}\n\\label{sec:Fz}\n\nThe $F$ factor depends on a cluster's angular size, which in turn\ndepends on the cluster's redshift.  We model the dependence of\n$F(\\theta_b, s,p)$ (Eq.~\\ref{eq:F}) on redshift by defining:\n\\begin{equation}\n\\label{eq:Fz}\nF_{\\mathrm m}(\\theta_b, s,\\beta,\\theta_c) =\n1 - \\frac{\\int b(\\mathbf{\\hat n},\\theta_b,s) \\, I_\\mathrm{SZ}(\\mathbf{\\hat n},\\beta,\\theta_c)\\d\\Omega}{\\int b(\\mathbf{\\hat n},\\theta_b,s=0) \\, I_\\mathrm{SZ}(\\mathbf{\\hat n},\\beta,\\theta_c)\\d\\Omega},\n\\end{equation}\nwhere $b(\\mathbf{\\hat n},\\theta_b,s)$ is a Gaussian beam profile\nwith beam width $\\theta_b$ , offset by angular distance $s$ from\nthe cluster center direction $\\mathbf{\\hat n_0}$.  We choose\n$s=s_{\\rm OCRA}$ to simulate the position of the OCRA reference\nbeam when the primary beam points at the cluster center.\n$I_\\mathrm{SZ}(\\mathbf{\\hat n},\\beta,\\theta_c)$ is a normalized,\nLOS integrated $\\beta$ profile that represents the SZ effect\nsurface brightness:\n\\begin{equation}\n\\label{eq:beta}\nI_\\mathrm{SZ}(\\mathbf{\\hat n},\\beta, \\theta_c) \\propto\n\\left(1+ \\frac{\\theta^2}{\\theta_c^2} \\right)^{\\frac{1}{2}-\\frac{3}{2}\\beta},\n\\end{equation}\nwhere $\\theta_c=2 r_c\/d_A(z)$ is the angular diameter of the\nobserved galaxy cluster defined in terms of its core size $r_c$,\n$\\theta$ is the angle from $\\mathbf{\\hat n}$ to $\\mathbf{\\hat\nn_0}$, and $d_A(z)$ is the angular diameter distance.\n$F_{\\mathrm m}$ depends on the choice of cosmological parameters\nand on the chosen cluster density profile. We calculate\n$F_{\\mathrm m}$ for $\\Lambda$CDM cosmological parameters:\n$h=0.7$, $\\Omega_m=0.3$, $\\Omega_\\Lambda=0.7$, and for an\nEinstein--de~Sitter cosmological model. For our redshift range we\nfind that the dependence on cosmological parameters is weak\ncompared to the dependence on the halo density profile\n(Fig.~\\ref{fig:f_z}).  We also calculate $F_{\\mathrm m}$ for the\ncase of a Gaussian halo but find that such profile is strongly\ndisfavored by simulations as $F_{\\mathrm m}$ approaches unity at\nfairly low redshifts.\n\n\nThe simplest $\\beta$-model does not allow for the steepening of\ndensity profiles with increasing $\\theta$.  However, X-ray\nobservations suggest that such steepening is real\n(e.g. \\citealt{Vikhlinin2006}), and it is expected that at large\ndistances from cluster cores (or higher redshifts) the\n$\\beta$-model yields lower $F_{\\mathrm m}(z)$ values than those\npredicted by simulations, as seen in Fig.~\\ref{fig:f_z}.\n\nClearly, the \\sampleHS sample has a large scatter in $F$ values\nat high redshifts. Some of that scatter is due to projection\neffects, which we discuss latter.  Heavy clusters of the\n\\sampleHS sample appear more compatible with the $\\beta$-model at\nlower $\\beta$ values than the lower-mass clusters of the\n\\sampleDF sample.  At the OCRA beam separation, the low-mass\nclusters in both samples show very weak effects of biasing\n($F\\approx 1$) at the highest redshifts.  On the other hand heavy\nhalos require large corrections, some of which do not result from\nsimple projection effects. In the next section, a selection of\nhalos are investigated individually.\n\n\n\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{figure_10.eps}\n\\includegraphics[width=0.49\\textwidth]{figure_11.eps}\n\\caption{Simulated fractions ($F$) as a function of redshift and\nvirial mass for clusters from \\sampleDF (left) and \\sampleHS\n(right) samples and for observations at effective beams angular\nseparation ${\\protect s=0.0526^\\circ}$ (the separation of OCRA\nbeams).  The fractions were measured from maps containing tSZ\nsignal only.  The lines trace the dependence for a halo\ndescribed by a $\\beta$-model according to Eq.~\\ref{eq:Fz} with\nparameters given in the plot legend.  The rectangles ``1'' and\n``2'' mark strong outliers and some halos from the main group\nthat are inspected individually (see text for discussion).  }\n\\label{fig:f_z}\n\\end{figure*}\n\n\n\\subsection{Analysis of individual clusters}\n\n\\begin{table}[t]\n\\caption{Parameters of halos selected from Fig.~\\ref{fig:f_z}.\nThe parameter $y_0$ is the maximal value of the LOS integrated\ncomptonization parameter.  }\n\\begin{ruledtabular}\n\\begin{tabular}{lccccc}\n\nID & $z$ & $F$ & $M_{\\rm vir}$ & $y_0\\times 10^5$ & comment\\footnotemark[1]\\\\\n& & & $[10^{14} M_\\odot\/h]$  &&\\\\\n\\hline\n\\multicolumn{6}{c}{\\sampleHS sample (rect. ``1'' selection)}\\\\\n1  & 0.552 & 0.39 & 4.6  & 1.38 & P \\\\\n2  & 0.520 & 0.43 & 4.7  & 3.40 & P \\\\\n3  & 0.885 & 0.49 & 4.4  & 0.71 & P, D \\\\\n4  & 0.429 & 0.47 & 4.6  & 2.10 & P, E \\\\\n5  & 0.366 & 0.44 & 5.1  & 1.75 & P \\\\\n6  & 0.370 & 0.53 & 4.9  & 2.10 & P \\\\\n7  & 0.349 & 0.50 & 5.1  & 3.10 & P \\\\\n8  & 0.337 & 0.34 & 5.4  & 3.70 & P \\\\\n9  & 0.388 & 0.50 & 4.4  & 2.2  & P \\\\\n\\multicolumn{6}{c}{\\sampleHS sample ($M_{\\rm vir}>12.7\\times 10^{14} M_\\odot\/h$)}\\\\\n10 & 0.103 & 0.41 & 13.8 & 11.2 & D \\\\\n11 & 0.109 & 0.56 & 14.6 & 25.6 & R, P \\\\\n12 & 0.193 & 0.49 & 12.8 & 5.6  & E, D, S, P\\\\\n\\multicolumn{6}{c}{\\sampleDF sample (rect. ``1'' selection)}\\\\\n13 & 0.367 & 0.44 & 2.1  & 0.93  & P \\\\\n\\multicolumn{6}{c}{\\sampleHS sample (rect. ``2'' selection)}\\\\\n14 & 0.182 & 0.64 & 11.3 & 6.68  & D \\\\\n15 & 0.268 & 0.77 & 12.6 & 9.50  & D \\\\\n16 & 0.169 & 0.75 & 10.1 & 10.3  & R, S\\\\\n\n\\end{tabular}\n\\end{ruledtabular}\n\\footnotetext[1]{P - reference beam flux density contamination\nfrom another halo due to LOS projection; D - disturbed\nmorphology; E - elongated shape; R - regular morphology\n(virialized halo); S - sub-halo(s) present;}\n\\label{tab:halos}\n\\end{table}\n\nIn Fig.~\\ref{fig:f_z} some of the halos are selected by\nrectangles in the $z-F$ diagram.  The properties of some of these\nhalos are given in Table.~\\ref{tab:halos}.  Fig.~\\ref{fig:f_z}\nshows that only the lightest halos in our samples are found to be\nstrong outliers, which is unsurprising.\n\nWe visually inspected all the clusters listed in\nTable~\\ref{tab:halos} and verified that each of the clusters from\nrectangle ``1'' (halo IDs from 1 to 9, and 13) lie at sky\npositions that are partially within another cluster's atmosphere\nand also within the angular distance of the reference beam. An\nexample of such overlap is shown in Fig.~\\ref{fig:halos} (top\npanels).\n\nInspection of the three highest mass clusters in the \\sampleHS\nsample (halo IDs 10, 11, and 12; black dots in right panel of\nFig.~\\ref{fig:f_z}) show that two of them (IDs 11 and 12) are\nalso affected to some degree by a LOS projection, but the\nmorphology of halo 10 shows no signs of another halo in the\ncomposite high-resolution map.  Instead, the SZ signature has a\ndisturbed morphology with angular extents larger than a single\nOCRA beam separation even though all three are at redshift\n$z>0.1$. This results in small $F$ values, and motivates\nmeasurements at larger angular separations.\n\nIn order to test whether high redshift clusters that\nsignificantly contribute to the scatter in the $F$--$z$ plane\n(Fig.~\\ref{fig:f_z}) could also benefit from observations out to\nangular distances beyond $s_{\\rm OCRA}$, we investigate the three\nmost massive clusters from rectangle ``2'' (Fig.~\\ref{fig:f_z},\nIDs: 14,15 and 16).  Their corresponding $F$ values\n(Tab.~\\ref{tab:halos}) do not seem to result from projection\neffects. Instead, these clusters have extended atmospheres and\/or\nstrongly disturbed and asymmetric SZ profiles (e.g. cluster 14,\nFig.~\\ref{fig:halos2}).\n\n\nSome of the heavy clusters have surface brightness profiles\n(Fig.~\\ref{fig:halos2}) that are strongly inconsistent with an\naxially-symmetric $\\beta$-profile. This necessitates using more\nsophisticated two-dimensional profiles at the data analysis stage\n\\citep{Lancaster2011,Mirakhor2016}.  Clearly, heavy halos\ngenerate low $F$ values and require large flux density\ncorrections with an OCRA type standard observational strategy\n(Sec.~\\ref{sec:strategy}).  These low $F$ values may partially\nstem from spurious projection effects (e.g. halos 11 and 12)\nwhich arise at the FOV generation stage for halos from the\n\\sampleHS sample (see Sec.~\\ref{sec:discussion}).\n\nThe outlying halos (rectangle ``1'') are either mergers (close\npairs of SZ-strong halos), or have elongated of disturbed\nmorphology (e.g. halos 3 and 4), or have small scale\nsub-structures.  However, in many cases these properties occur\nat spatial scales that will not be resolved in OCRA~SZ\nobservations and\/or may be relevant only as galaxy scale SZ\neffects that are too faint to be detected.\n\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=\\haloPlotWidth\\textwidth]{figure_12.eps}\n\\includegraphics[width=\\haloPlotWidth\\textwidth]{figure_13.eps}\n\\includegraphics[width=\\haloPlotWidth\\textwidth]{figure_14.eps}\n\\includegraphics[width=\\haloPlotWidth\\textwidth]{figure_15.eps}\n\\includegraphics[width=\\haloPlotWidth\\textwidth]{figure_16.eps}\n\\includegraphics[width=\\haloPlotWidth\\textwidth]{figure_17.eps}\n\\includegraphics[width=\\haloPlotWidth\\textwidth]{figure_18.eps}\n\\includegraphics[width=\\haloPlotWidth\\textwidth]{figure_19.eps}\n\\caption{Selection of simulated Compton $y$-parameter profiles\n(in arbitrary units) for halos from Table.~\\ref{tab:halos}. The\npanels show profiles for individual halos in physical\ncoordinate space ({\\em left}), and their coarse-grained version\nobtained from high resolution maps in angular space with\ncontributions from other halos along the LOS ({\\em right}).\nThe position of halos in the left-hand side panels is defined\nby a box size that contains all FOF particles of the halo\nassociated with a given cluster. In the right-hand side panels\nthe SZ peak for the cluster is located in the plot center. For\nany given cluster the flux density calculation is done at the\nsky position of the peak. For each cluster the black circles\ndenote OCRA FWHMs and their relative separation ($s_{\\rm\nOCRA}$).  }\n\\label{fig:halos}\n\\end{figure}\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=\\haloPlotWidth\\textwidth]{figure_20.eps}\n\\includegraphics[width=\\haloPlotWidth\\textwidth]{figure_21.eps}\n\\includegraphics[width=\\haloPlotWidth\\textwidth]{figure_22.eps}\n\\includegraphics[width=\\haloPlotWidth\\textwidth]{figure_23.eps}\n\\includegraphics[width=\\haloPlotWidth\\textwidth]{figure_24.eps}\n\\includegraphics[width=\\haloPlotWidth\\textwidth]{figure_25.eps}\n\\caption{As in Fig.~\\ref{fig:halos} but for the selection of halos from rectangle ``2''. See Table.~\\ref{tab:halos} for details.}\n\\label{fig:halos2}\n\\end{figure}\n\n\n\n\n\\subsection{Practical aspects of using CMB templates}\n\\label{sec:template}\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.47\\textwidth]{figure_26.eps}\n\\caption{Residual histogram (i.e. observational pixel\nfrequencies minus median pixel frequencies estimated from an\nensemble of Gaussian NILC map simulations) of the CMB\ntemperature fluctuations at and around the {\\em Planck} SZ\ngalaxy clusters, measured in the {\\em Planck} NILC inside\ncircular apertures of radius $r$ centered at the clusters'\npositions (solid); and $1\\sigma$, $2\\sigma$ and $3\\sigma$\nconfidence contours of the pixel frequencies in these\nsimulations (dashed).}\n\\label{fig:NILC-test}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.47\\textwidth]{figure_27.eps}\n\\caption{As in Fig.~\\ref{fig:NILC-test} but for the {\\em Planck}\n217~GHz frequency map and only for clusters at galactic\nlatitude $b>60^\\circ$.}\n\\label{fig:217GHz-test}\n\\end{figure}\n\n\nBy subtracting the templated version of the CMB map\n(Sec.~\\ref{sec:CMBsims}) from the pure CMB simulation, it is easy\nto estimate the upper limit of the residual CMB signal captured\nin the OCRA difference beam observations.  The {\\em Planck}\n217~GHz and NILC maps have enough pixels to create a template of\nresolution of the order of an arcminute, and at least the former\nshould contain only a negligible tSZ signal.\n\nAlthough subtracting CMB templates from the pure CMB maps\ndecreases the large-scale variance by an order of magnitude, the\nresidual variance in the map is carried by high frequency noise\nthat will generate a small amount of dispersion in difference\nobservations (Fig.~\\ref{fig:templates} bottom-right panel).\nHowever, the residual small-scale noise should approximately\naverage out under rotation of the beams in the sky, and since the\nlarge-scale power is effectively removed, increasing the\neffective separation should not suffer from exponential variance\ngrowth due to primordial CMB at arcminute angular scales.\n\nHow reliable are the 217~GHz or NILC {\\em Planck} templates in\ncorrecting single frequency SZ observations for confusion with\nthe primordial CMB?  The 217~GHz map is foreground contaminated\nand the NILC map, although foreground cleaned, still may contain\nresidual tSZ signals at scales least optimized in the needlet\nspace.\n\nIn order to quantify the foregrounds and residual tSZ\ncontaminaiton in each map, we calculate histograms of the\ntemperature fluctuation distribution outside of a mask that\nremoves the full sky except for the directions towards {\\em\nPlanck}-detected galaxy clusters from the PCSS SZ union R.2.08\ncatalog \\citep{PlanckCollaboration2015f}.  Each non-masked region\nis a circular patch of radius $a=\\{2.5',5',10',15'\\}$.\nForegrounds will generate strong positive skewness in the\ntemperature distribution, while the presence of residual tSZ in\nthe NILC map should manifest itself by either a positive or\nnegative skew depending on the frequency weights in the internal\nlinear combination.\n\nWhile the results of the test for the {\\em Planck} NILC map\n(Fig.~\\ref{fig:NILC-test}) do not give strong deviations from\nGaussian simulations, the 217~GHz map generally does.  The data\nare inconsistent with Gaussian simulations even at high galactic\nlatitudes (Fig.~\\ref{fig:217GHz-test}), although the significance\nof the foregrounds seems to depend on the size of the circular\npatch. This implies that the 217~GHz frequency map cannot readily\nbe used to mitigate the confusion due to CMB in OCRA observations\nwithout further assumptions on the foregrounds' frequency\ndependence.  However, it should be interesting to quantify the\nsignificance of the arcminute scale Galactic foregrounds at\n30~GHz at high and intermediate latitudes for OCRA difference\nobservations with small beam separations.  {\\em Planck}-LFI data\nmight also help reduce these foregrounds, though we do not study\nthis here.\n\nSince the foreground cleaned NILC map is statistically consistent\nwith Gaussian simulations\n\\citep{PlanckCollaboration2015a,PlanckCollaboration2015d} towards\nthe {\\em Planck}-detected galaxy clusters\n(Fig.~\\ref{fig:NILC-test}), it should also be suitable for\nmitigating CMB confusion in OCRA observations in directions\noutside of the mask where clusters undetected by {\\em Planck}\nlie.\n\n\\subsection{Pointing requirements}\n\\label{sec:pointing}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{figure_28.eps}\n\\includegraphics[width=0.45\\textwidth]{figure_29.eps}\n\\caption{Systematic effects in tSZ flux density reconstruction\nfrom dual-beam observations as a function of telescope pointing\nerrors ($\\epsilon_p$) and beam separation $s$ for the\n\\sampleDF~sample (left) and for the \\sampleHS~sample (right).\nThe reference beam is assumed to cover all possible parallactic\nangles for any given galaxy cluster.  }\n\\label{fig:F_vs_pointing}\n\\end{figure*}\n\nIn order to quantify the implications of telescope pointing\nerrors on the reconstruction of Compton $y$-parameters, and to\ndefine pointing requirements, we introduce a pointing precision\nparameter $\\epsilon_p$ that defines the maximal angular distance\nthat a primary beam can have from the intended position, and then\nwe repeat the analysis of Sec.~\\ref{sec:beam_separation}.  The\npointing error $p$ is drawn from a uniform distribution on\n$[0,\\epsilon_p]$, since the RT32 pointing and tracking are\ndominated by systematic errors, and we investigate different\nvalues of $\\epsilon_p$.  Since galaxy cluster SZ profiles are\ntypically steep functions of angular separation, any pointing\ninaccuracy will lead to biasing measurements of the central\ncomptonization parameter when taking averages from multiple\nobservational sequences.\n\nFigure~\\ref{fig:F_vs_pointing} shows that for the \\sampleHS\nsample, i.e. typically heavy clusters, pointing error up to\n$\\epsilon_p\\approx {\\rm HPBW}\/4$ should not lead to strong\n($>10\\%$) extra biases relative to the $\\epsilon_p=0$ case.\nMeasurements of the \\sampleDF sample, i.e. typically less massive\nclusters, are more sensitive to pointing errors, but if the\npointing accuracy is better than $\\epsilon_p=0.005^\\circ$ (${\\rm\n\\theta_b^{\\rm OCRA}}\\approx 1.2'$) the additional systematic\neffects will be smaller than $10\\%$.  However, larger pointing\nerrors should be taken into account at the data analysis stage. An\nobservational campaign is currently under way to improve RT32 pointing\naccuracy.\n\n\\section{Discussion}\n\\label{sec:discussion}\n\nThe map-making procedure that has been tested for recovering the\nsource intensity distribution from OCRA difference measurements\n($s=s_{\\rm OCRA}$) assumes a flat background. This is not a\nproblem for reconstructions of comptonization parameters from SZ\nobservations of heavy clusters, as with the standard OCRA\nbeam-pair separation the corrections due to CMB background are\nsmall.  However, reconstructing cluster SZ profiles out to larger\nangular distances could benefit from correcting the difference\nmeasurements according to the {\\em Planck} CMB template.  For\nexample, Fig.~\\ref{fig:medianfHS} (left panel) shows that an\nobservation at $s=2 s_{\\rm OCRA}$ decreases bias by $\\Delta\nF\\approx 0.2$. At the same time CMB confusion broadens the 68\\%\nCR by $\\Delta F \\approx 0.1$, but applying a {\\em Planck} CMB\ntemplate reverses this effect almost down to the intrinsic\ntSZ+kSZ scatter.\n\nAs discussed in Sec.~\\ref{sec:sample} the \\sampleDF and \\sampleHS\nsamples represent quite opposite observational\napproaches. However, since halos of the \\sampleHS sample were\nselected from full simulation volumes (rather than from\nlight-cone sections), mock maps for this sample contain clusters\nwith angular sizes calculated according to their redshifts and\nphysical extents, as in the case of FOV simulations, but are\nplaced in the map at rectilinearly projected locations.  This\ncontaminates the resulting maps with halos that would not fall\ninto the assumed FOV in the standard light-cone approach.  These\nspurious halo--halo overlaps may somewhat enlarge the 68\\% CR\ncontours of various $F$ distributions (e.g. x=tSZ or x=tSZ+CMB).\nA possible modification of the calculation scheme for the\n\\sampleHS sample would be to consider each halo independently,\nthus completely ignoring the intrinsic projection effects that\nexist in the light-cone approach, or by extending the FOV to a\nhemisphere (which would probably require implementing adaptive\nresolution maps to maintain the angular resolution of the present\ncalculations).\n\nThe cluster samples that we analyze were not screened to select\nvirialized clusters.  Although we analyzed sub-samples selected\nusing a virialization criterion (based on ratios of potential to\nkinetic energy of FOF halo particles) the results presented here\nare based on the full sample in order to retain a morphological\nvariety of SZ galaxy cluster profiles (Figs.~\\ref{fig:halos} and\n~\\ref{fig:halos2}), and to expose the complexity of SZ flux\ndensity reconstructions from observations that do not intend to\ncreate multi-pixel intensity maps.\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nWe quantify the significance of systematic effects arising in\ndual-beam, differential observations of Sunyaev-Zel'dovich (SZ)\neffect in galaxy clusters.  We primarily focus on effects\nrelevant to the reconstruction of comptonization parameters from\nsingle frequency flux-density observations performed with the One\nCentimeter Receiver Array (OCRA) -- a focal plane receiver with\narcminute scale beamwidths and arcminute scale beam separations\n-- installed on the 32~m radio telescope in Toru\\'n.\n\nUsing numerical simulations of large scale structure formation we\ngenerate mock cluster samples (i) from blind surveys in small\nfields of view and (ii) from volume limited targeted observations\nof the most massive clusters (Sec.~\\ref{sec:sample}).  Using mock\nintensity maps of SZ effects we compare the true and recovered SZ\nflux densities and quantify systematic effects caused by the\nsmall beam separation, by primary CMB confusion and by\ntelescope pointing accuracy.\n\nWe find that for massive clusters the primary CMB confusion does\nnot significantly affect the recovered SZ effect flux density\nwith OCRA beam angular separation of $\\approx 3'$. However, these\nobservations require large corrections due to the differential\nobserving strategy. On the other hand, measurements of SZ-faint\n(or high redshift $0.4<z<1.0$) clusters may have their SZ\nphotometry erroneously estimated by $10\\%$ or more due to CMB\nconfusion, which becomes stronger in observations that map larger\nangular distances from cluster centers.\n\nWe investigate the possibility of mitigating the CMB confusion in\nSZ observations that map scales beyond $3'$ from cluster centers\nby using {\\em Planck} CMB 217~GHz and foreground reduced NILC\nmaps as primary CMB templates. Using simulations we find that\nthese templates have sufficiently high angular resolution\/low\nnoise to significantly mitigate CMB confusion in 30~GHz\nobservations of high-$z$ clusters, given that the templates do\nnot contain residual foregrounds. Using a simple one-point\nstatistic, targeted towards directions of known clusters, we\nverify that at least the {\\em Planck} NILC map should also be\nsufficiently free from foregrounds to serve as a primary CMB\ntemplate (Sec.~\\ref{sec:template}) that could improve OCRA-SZ or\nsimilar observations extended to larger angular scales\n(Sec.~\\ref{sec:strategy}).\n\nFinally, we find that RT32 telescope pointing and tracking\naccuracy $\\epsilon_p\\ < 0.005^\\circ$ should keep systematic\nerrors in recovered SZ flux densities (comptonization parameters)\nbelow $\\approx 5\\%$ (after correcting for other systematic\neffects) even in observations of SZ-weak clusters\n(Sec.~\\ref{sec:pointing}).\n\n\\section*{Acknowledgments}\nThank you to Mark Birkinshaw for discussion on OCRA observational\nstrategies, and to an anonymous referee for useful comments.\nThis research has made use of a modified version of the\nGPL-licensed ccSHT library.  We also acknowledge use of the\nmatplotlib plotting library \\citep{Hunter2007}.\nThis work was financially supported by the Polish National\nScience Centre through grant DEC-2011\/03\/D\/ST9\/03373.  A part of\nthis project has made use of computations made under grant 197 of\nthe Pozna{\\'n} Supercomputing and Networking Center (PSNC).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}