{"id":5609,"date":"2012-02-14T22:23:48","date_gmt":"2012-02-14T22:23:48","guid":{"rendered":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/?p=5609"},"modified":"2012-02-14T22:23:48","modified_gmt":"2012-02-14T22:23:48","slug":"5609","status":"publish","type":"post","link":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/?p=5609","title":{"rendered":""},"content":{"rendered":"<p>I was reading an <a href=\"http:\/\/arxiv.org\/abs\/math\/0506416\" title=\"amusing paper\" target=\"_blank\">amusing paper<\/a> of Ronald van Luijk: he demonstrates<br \/>\nthe K3 surface with Picard number one.<\/p>\n<p>Reflecting his proof I formulate the following unexpected principle:<\/p>\n<p>in order to classify Fanos with Picard number 1<br \/>\nwe should NOT just get rid of all polytopes with higher Picard number !<\/p>\n<p>One of the reasons is as follows:<br \/>\n0. Assume we&#8217;ve done with computing basic invariants of polytopes,<br \/>\nincluding Picard numbers<br \/>\n1. Assume we&#8217;ve done with the initial step of creating some database<br \/>\nof prospective Laurent polynomials out of many many polytopes<br \/>\n2. It is much easier to create a &#8220;phone book&#8221; of period sequences (say<br \/>\nfirst 10 or 20 coefficients) then to compute their Picard&#8211;Fuchs<br \/>\noperators.<br \/>\n3. Once we have a &#8220;phone book&#8221; prospective polynomials W fall into<br \/>\nequivalence classes with respect to initial terms of their period<br \/>\nsequences<br \/>\n   If we manage  how to program Cremona-equivalence then they fall<br \/>\ninto even better classes.<br \/>\n4. Now, under assumption that Laurent polynomial W reflects a toric<br \/>\ndegeneration of Fano manifold X into the toric variety X_0 that W is<br \/>\nsupported on (i.e. Newton(W) is dual to Moment(X_0))<br \/>\n   $ rk Pic X \\geq \\rk \\Pic X_0 $<br \/>\n5. So if we know that period sequence is supported on some toric<br \/>\nvariety with Picard number > 1 &#8211; then it is not a period sequence of<br \/>\nFano with Picard number 1<br \/>\n   If we would drop polytopes with Picard number >1, then we have to<br \/>\ndo extra work!<\/p>\n<p>In fact we used this ideas for threefolds before we computed their<br \/>\nG-series by ad hoc and a posteriori methods.<br \/>\nBy a posteriori I mean it is not so easy to tell what is Picard number<br \/>\nof Fano variety from its G-series and\/or differential operator that<br \/>\nannihilates it:<br \/>\nG-series for Fano varieties with Picard number 1 and for G-minimal<br \/>\n(hence quantum minimal) varieties look pretty much the same.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>I was reading an amusing paper of Ronald van Luijk: he demonstrates the K3 surface with Picard number one. Reflecting his proof I formulate the following unexpected principle: in order to classify Fanos with Picard number 1 we should NOT just get rid of all polytopes with higher Picard number ! One of the reasons [&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":[],"class_list":["post-5609","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts\/5609","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=5609"}],"version-history":[{"count":3,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts\/5609\/revisions"}],"predecessor-version":[{"id":5612,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts\/5609\/revisions\/5612"}],"wp:attachment":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5609"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5609"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5609"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}