{"id":5638,"date":"2012-02-25T13:03:27","date_gmt":"2012-02-25T13:03:27","guid":{"rendered":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/?p=5638"},"modified":"2012-02-25T13:15:00","modified_gmt":"2012-02-25T13:15:00","slug":"5638","status":"publish","type":"post","link":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/?p=5638","title":{"rendered":"Picard lattices of Fano threefolds"},"content":{"rendered":"<p><a href=\"http:\/\/sergey.ipmu.jp\/papers\/NodalToricFanoThreefoldPicard.gp\">The updated script for computing Picard lattices of Fano threefolds: now it works and, even better, computes all five principal invariants of the smoothing!<\/a><\/p>\n<p><a href=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2012\/02\/NodalToric3foldPicard1.pdf\">Picard lattices of ambiguous nodal toric Fano threefolds<\/a><\/p>\n<p>Let <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/021\/02129bb861061d1a052c592e2dc6b383-ffffff-000000-0.png' alt='X' title='X' class='latex' \/> be a nodal toric Fano threefold (recall that in toric world terminal Gorenstein singularities of Fano threefolds are simply ordinary double points aka nodes <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/232\/2322caf126e6be2e67d7eb088fd671e9-ffffff-000000-0.png' alt='(xy=zt) \\subset \\CC^4 = \\mathrm{Spec } \\CC[x,y,z,t]' title='(xy=zt) \\subset \\CC^4 = \\mathrm{Spec } \\CC[x,y,z,t]' class='latex' \/> ).<\/p>\n<p>Given a terminal Gorenstein toric Fano threefold <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/021\/02129bb861061d1a052c592e2dc6b383-ffffff-000000-0.png' alt='X' title='X' class='latex' \/>,<\/p>\n<p>this script do the following:<\/p>\n<p>1. Compute Picard lattice <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/150\/1505cb2029ded0cf9fd0d7f5cb1ae906-ffffff-000000-0.png' alt='Pic(X)' title='Pic(X)' class='latex' \/><br \/>\n2. Then compute (self)intersection theory on this lattice.<br \/>\nThis part is done in 3 steps:<br \/>\na. pick a small crepant resolution <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/65d\/65dc53d4cd181c30d3d9f0c0a0ddee28-ffffff-000000-0.png' alt='\\pi : \\hat{X} \\rightarrow X' title='\\pi : \\hat{X} \\rightarrow X' class='latex' \/><br \/>\nb. compute intersection theory of smooth toric manifold <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/fd2\/fd2b1eb3a3aacb801a8c6d0b7ec448b5-ffffff-000000-0.png' alt='\\hat{X}' title='\\hat{X}' class='latex' \/>,<br \/>\nc. restrict intersection theory from <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/0d4\/0d48daa68c75f33250071135d50eac32-ffffff-000000-0.png' alt='Pic \\hat{X}' title='Pic \\hat{X}' class='latex' \/> to <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/4cc\/4cc4686377becf7fa8704ebc1f29be1f-ffffff-000000-0.png' alt='\\pi^* Pic X \\cong Pic X' title='\\pi^* Pic X \\cong Pic X' class='latex' \/>.<\/p>\n<p>3. Threefold X has a unique deformation class of smoothing by Fano threefold Y<br \/>\nWe also compute the principal invariants of Y: Betti numbers, degree, Lefschetz discriminant and Fano index<\/p>\n<p>The main procedure is called Picard(toric)<br \/>\nThe input is a 3-component vector toric=[description, vertices, faces]<br \/>\n description is a verbal description of variety X (not used for computations)<br \/>\n vertices is a matrix of vertices of the fan polytope Delta(X)<br \/>\n faces is a transposed matrix of faces (vertices of the moment polytope)<\/p>\n<p>The output is a 2-component vector o = [lattice, invariants]<br \/>\n where lattice is 3-component vector [cubic, M, class]<br \/>\n  cubic is homogenous cubic polynomial of &#8216;rk Pic(X)&#8217; variables (self-intersection pairing)<br \/>\n  class is the expression of the first Chern class in terms of generators of Picard group<br \/>\n  M is the matrix of the Lefschetz pairing <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/bc1\/bc107c1ee4b059ffa052e7bdad6cce49-ffffff-000000-0.png' alt='(a,b) \\to \\int_{[X]} a \\cup b \\cup c_1(X) ' title='(a,b) \\to \\int_{[X]} a \\cup b \\cup c_1(X) ' class='latex' \/><br \/>\n and invariants is 5-component vector [rho,deg,b,d,r] of the principal invariants<br \/>\n  rho is Picard number i.e. second Betti number of X<br \/>\n  deg is the anticanonical degree <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/ccc\/cccf259f6d55f73d0b09adfa44414f00-ffffff-000000-0.png' alt='\\int_{[X]} c_1(X)^3' title='\\int_{[X]} c_1(X)^3' class='latex' \/><br \/>\n  b is the half of third Betti number of the smoothing Y<br \/>\n  d is the Lefschetz discriminant (i.e. determinant of matrix M)<br \/>\n  r is the Fano index (i.e. divisibility of <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/4f1\/4f14b706ff5b5ebe58c603d14356e910-ffffff-000000-0.png' alt='c_1(X)' title='c_1(X)' class='latex' \/> in <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/fb5\/fb5f6d1d92365fe8558cf64c5479d5d2-ffffff-000000-0.png' alt='H^2(X,\\ZZ)' title='H^2(X,\\ZZ)' class='latex' \/>)<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The updated script for computing Picard lattices of Fano threefolds: now it works and, even better, computes all five principal invariants of the smoothing! Picard lattices of ambiguous nodal toric Fano threefolds Let be a nodal toric Fano threefold (recall that in toric world terminal Gorenstein singularities of Fano threefolds are simply ordinary double points [&hellip;]<\/p>\n","protected":false},"author":4,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[15,9,21,3,20],"class_list":["post-5638","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-code","tag-geometry","tag-k3","tag-theory","tag-threefolds"],"_links":{"self":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts\/5638","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5638"}],"version-history":[{"count":12,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts\/5638\/revisions"}],"predecessor-version":[{"id":5652,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts\/5638\/revisions\/5652"}],"wp:attachment":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5638"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5638"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5638"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}