{"id":256,"date":"2010-09-07T11:43:13","date_gmt":"2010-09-07T11:43:13","guid":{"rendered":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/?p=256"},"modified":"2011-06-15T03:49:52","modified_gmt":"2011-06-15T03:49:52","slug":"expected-distribution-of-equations","status":"publish","type":"post","link":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/?p=256","title":{"rendered":"Expected distribution of equations."},"content":{"rendered":"

Let X be a smooth Fano threefold with Picard number P = \\rho = dim H^2(X).<\/p>\n

Then subring of algebraic (even) cycles in X is (2+2P)-dimensional, and its Lefschetz decomposition has P blocks: 1 block of length 4 and (P-1) blocks of length 2. So its image in cohomologies of anticanonical section (K3 surface) is (2+2P – P) = (2+P)-dimensional.<\/p>\n

For “general” Fano threefold with Picard number P we expect
\nregularized quantum differential equation (RQDE) to be of degree (2+P) in D = t \\frac{d}{dt}
\nand to have (2+2P) singular points. Nevertheless degree in t may be more than number of singular points
\ndue to apparent singularities.<\/p>\n

It turns out that condition for general is not very general in practice.<\/p>\n

Assume Fano threefold X has action of finite group G in one of the 4 ways:
\na. G acts on X by regular (algebraic) transformations,
\nb. G acts on X by symplectic transformations,
\nc. X is defined over non-algebraically closed field k and G is Galois group Gal(k),
\nd. X is a fiber of a smooth family over some base B and fundamental group G = \\pi_1(B) acts on H^\\bullet(X) via monodromy. <\/p>\n

For cases a,b,c consider the induced action of G on cohomology of X.<\/p>\n

Let p = \\rho^G = dim H^2(X)^G be $G$-invariant Picard number.
\nG-invariant part of cohomologies H(X)^G is (2+2p)-dimensional.<\/p>\n

Define \\emph{minimal quantum cohomology subring } QH_m(X) of \\emph{very small quantum cohomology ring} QH(X) as subring generated by c_1(X) and C[t].
\nIt is easy to see QH_m(X) is contained in H(X, C)^G [t].<\/p>\n

This implies that regularized I-series I_X is annihilated by
\ndifferential operator of degree (2+p).<\/p>\n

So it is natural to ask about possible G-actions on Fano threefolds.
\nFirst (numerical) step is to see the possible automorphisms of Mori cone or Kaehler cone.
\nWe have some structures on H^2(X,R):
\na. lattice H^2(X,Z) and element c_1(X) inside the lattice,
\nb. rational polyhedral cone of numerically effective divisors,
\nc. nondegenerate integral quadratic form (Lefschetz pairing) : (A,B) -> \\int_X A \\cup B \\cup c_1(X).
\nWe call this information \\emph{Mori structure}.<\/p>\n

Group of automorphisms of Mori structure is finite, and for any action
\nG-invariant Picard number is not less than dimension of invariants of H^2(X)
\nwith respect to whole group of automorphisms of Mori structure.<\/p>\n

As far as I remember (but cannot find a reference) for all Fano threefolds one may find some moduli
\nand some kind of G-action such that G-invariant Picard group coincides with invariant part of $H^2$ with respect to automorphisms of Mori structure.<\/p>\n

The standard reference for automorphisms of Mori structure is probably:
\nKenji Matsuki, “Weyl groups and birational transformations among minimal models”, AMS 1995<\/p>\n

He studies slightly different problem, but has a similar answer. Unfortunately I haven’t a copy of this book, but copied one page from google books.<\/p>\n

He says automorphisms of Mori structures turn out to be Weyl groups.<\/p>\n

He claims the following Fano threefolds have nontrivial automorphisms:<\/p>\n

P – Picard number, then list of Mori-Mukai numbers with the given Picard number<\/p>\n

P=2:
\nA_1: 2, 6, 12, 21, 32 (these should be G-Fano, but number 2 is suspicious)
\nother have p=2<\/p>\n

P=3:
\nA_2: 1, 27 (G-Fano, suspicious that 13 is in the next line)
\nA_1: 3, 7, 9, 10, 13, 17, 19, 20, 25, 31 (should correspond to p=2)
\nother have p=3<\/p>\n

P=4:
\nA_3: 1 (G-Fano)
\nA_2: 6 (p=2)
\nA_1 \\times A_1: 2 (p=2)
\nA_1: 3, 4, 7, 8, 10, 12 (p=3)
\ntrivial – 5,9,11 (should have p=4)
\nmissing number 13 from Erratum<\/p>\n

P=5:
\nA_1 \\times A_2: 3 (p=2)
\nA_2: 1 (p=3)
\nA_1: 2 (p=4)<\/p>\n

For cases P \\geq 6 our threefolds are products of a line and del Pezzo surface P^1 \\times S_d. They all have Weyl group of type E_{9-d} and p=2.<\/p>\n

So the distribution in p is the following (case 4.13 is missing):
\np is always less than 5;
\np=4 – 4 varieties: 5.2; 4.5, 4.9, 4.11
\np=3 – 26 varieties: 5.1; 4.3, 4.4, 4.7, 4.8, 4.10, 4.12; and 19 with P=3
\np=2 – 50 varieties
\np=1 – 25 varieties (or 26 if 2.2 is there)<\/p>\n

This means just 4 varieties should have N=6, and other have even less.<\/p>\n

The obvious thing to do is to recompute ourselves the Mori structure and its automorphisms
\n(in particular discriminant of Lefschetz quadratic form is an important invariant that we need anyway).<\/p>\n","protected":false},"excerpt":{"rendered":"

Let X be a smooth Fano threefold with Picard number P = . Then subring of algebraic (even) cycles in X is (2+2P)-dimensional, and its Lefschetz decomposition has P blocks: 1 block of length 4 and (P-1) blocks of length 2. So its image in cohomologies of anticanonical section (K3 surface) is (2+2P – P) […]<\/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":[5,3],"class_list":["post-256","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-data","tag-theory"],"_links":{"self":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts\/256","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=256"}],"version-history":[{"count":14,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts\/256\/revisions"}],"predecessor-version":[{"id":266,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts\/256\/revisions\/266"}],"wp:attachment":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=256"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=256"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=256"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}