{"id":5195,"date":"2010-12-31T11:43:34","date_gmt":"2010-12-31T11:43:34","guid":{"rendered":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/?p=5195"},"modified":"2011-03-21T13:18:36","modified_gmt":"2011-03-21T13:18:36","slug":"a-riddle-wrapped-in-a-mystery-inside-a-balls-up","status":"publish","type":"post","link":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/?p=5195","title":{"rendered":"A riddle wrapped in a mystery inside a balls-up"},"content":{"rendered":"

Consider #2 on the Mori-Mukai list of rank-3 Fano 3-folds.\u00a0 This has been giving us some difficulty, which I have now resolved.\u00a0 We were making a combination of mistakes.\u00a0 Mori and Mukai describe the variety X as follows.<\/p>\n

A member of |L^{\\otimes 2} \\otimes_{\\cO_{\\PP^1 \\times \\PP^1}} \\cO(2,3)| on the \\PP^2-bundle \\PP(\\cO \\oplus \\cO(-1,-1)^{\\oplus 2}) over \\PP^1 \\times \\PP^1 such that X \\cap Y is irreducible, where L is the tautological line bundle and Y is a member of |L|. <\/strong><\/p>\n

Our first mistake, as Mukai-sensei pointed out in an email to Corti, was using the wrong weight convention for projective bundles.\u00a0 Mori and Mukai use negative weights, so the ambient \\PP^2-bundle F is the toric variety with weight data:
\n\\begin{array}{cccccccc} x_0 & x_1 & y_0 & y_1 & t_0 & t_1 & t_2 & \\\\ 1 & 1 & 0 & 0 & 0 & 1 & 1 & \\\\ 0& 0 & 1 & 1 & 0 & 1 & 1 & \\\\ 0 & 0 & 0 & 0& 1 &1 & 1 & \\end{array}
\nFor later convenience we change basis, expressing F as the toric variety with weight data:
\n\\begin{array}{cccccccc} x_0 & x_1 & y_0 & y_1 & t_0 & t_1 & t_2 & \\\\ 1 & 1 & 0 & 0 & -1 & 0 & 0 & A \\\\ 0& 0 & 1 & 1 & -1 & 0 & 0 & B \\\\ 0 & 0 & 0 & 0& 1 &1 & 1 & C \\end{array}
\nX is a section of |B+2C|.<\/p>\n

Our second mistake was failing to accurately account for the fact that although X is Fano, the bundle A+C (which restricts to -K_X on X) is only semi-positive<\/em> on F.\u00a0 Thus we are in the situation described in this post<\/a> and, in the notation defined there, we have:
\n\\begin{cases} F(q) = 1 \\\\ G(q) = 2q_3+6q_2 q_3 \\\\ H_1(q) = \\sum_{b>0} {(-1)^{b} \\over b} q_2^b = {-\\log(1+q_2)} \\\\ H_2(q) = {-\\log(1+q_2)} \\\\ H_3(q) = \\log(1+q_2) \\end{cases}
\nand hence:
\n\\begin{cases} q_1 = {\\hat{q}_1 \\over 1 - \\hat{q}_2} \\\\ q_2 = {\\hat{q_2} \\over 1 - \\hat{q_2}} \\\\ q_3 = \\hat{q}_3 (1-\\hat{q_2}) \\end{cases}
\nThus the cohomological-degree-zero part of the J-function is:
\n\\exp({-2}\\hat{q}_3(1-\\hat{q}_2)-6\\hat{q}_2 \\hat{q}_3) \\sum_{a,b,c\\geq 0} \\hat{q}_1^a \\hat{q}_2^b \\hat{q}_3^c(1-\\hat{q}_2)^{c-b-a} {1 \\over z^{a+c}} {(b+2c)! \\over a!a!b!b!c!c!(c-b-a)!}
\nand setting \\hat{q}_1 = t, \\hat{q}_2 = 1, \\hat{q}_3 = t, z=1 yields:
\n\\exp(-6t) \\sum_{a,b\\geq 0} t^{2a+b} {(2a+3b)! \\over a!a!b!b!(a+b)!(a+b)!}
\nRegularizing this gives:
\nI_{reg}(t) =1+58 t^2+600 t^3+13182 t^4+247440 t^5+5212300 t^6+111835920 t^7+ \\cdots
\nAs Galkin conjectured, this is period sequence 97.<\/p>\n","protected":false},"excerpt":{"rendered":"

Consider #2 on the Mori-Mukai list of rank-3 Fano 3-folds.\u00a0 This has been giving us some difficulty, which I have now resolved.\u00a0 We were making a combination of mistakes.\u00a0 Mori and Mukai describe the variety as follows. A member of on the -bundle over such that is irreducible, where is the tautological line bundle and […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[5,10,3],"class_list":["post-5195","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-data","tag-example","tag-theory"],"_links":{"self":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts\/5195","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\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5195"}],"version-history":[{"count":13,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts\/5195\/revisions"}],"predecessor-version":[{"id":5391,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts\/5195\/revisions\/5391"}],"wp:attachment":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5195"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5195"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5195"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}