Notations:
\np.s. – period sequence
\nfano p.s. – one of 105 p.s. of smooth Fano 3-folds
\ngood p.s. – p.s. s.t. P_0(D) has integer roots, where L = sum_{i=0}^r t^i P_i(D) is the diff.op. annihilating p.s.
\nbad p.s. – not good p.s.<\/p>\n
LP – Laurent polynomial.
\nfano LP – Laurent polynomial for which RG coincides with one of 105 fano p.s.<\/p>\n
Everywhere below we put a binomial condition (of Coates-Corti-Galkin-Golyshev-Przjyalkowski-Usnich-etc) on the edges.<\/p>\n
MP – Minkowski Laurent polynomial
\nPP – (Coates-)Przyjalkowski Laurent polynomial (binomial on edges, zero everywhere else)<\/p>\n
This is the statistics of reflexive 3-polytopes.<\/p>\n
—<\/p>\n
18 are smooth.<\/p>\n
—<\/p>\n
100 (= 18+82) are terminal. They give rise to 100 LP and 62 good p.s.<\/p>\n
—<\/p>\n
899 (= 18+82+799) has no integer points in interior of facets.<\/p>\n
Each one gives rise to a unique MP (which coincides with PP).<\/p>\n
712 of these MPs are fano LP – they give rise to 92 fano p.s.
\n187 are bad – they give rise to 63 bad p.s.<\/p>\n
92 = 98-6, where 98 = 105-7.
\nThe 7 that didn’t appear among MP p.s.’s are:
\nV_2 [bottom degree],
\nits double cover B_1 [bottom degree in rho=1, r=2],
\n2.1, 2.2, 2.3 [bottom degree in rho=2],
\nP^1 x S_2, P^1 x S_1 [top Picard rank].<\/p>\n
The extra 6 Fano (p.s.) that didn’t appear are:
\nV_4 (15), V_6 (19), V_8 (5), V_{10} (9), V_{12} (7) [next bottom degree in rho=1,r=1);
\n2.4 (49) [next bottom degree in rho=2].<\/p>\n
So they lie at the bottom of the list, just over the non-appearing 5.<\/p>\n
—<\/p>\n
1051 polytopes has exactly one integer point in the respective interior of the facets (i.e. not origin, vertex, and not on the edge).<\/p>\n
The facet that contains the extra integer point is then on of famous 16 reflexive 2-polytopes,
\nso 1051 polytopes fall into 16 classes.<\/p>\n
Number of polytopes in each class is as follows (total 1051):
\n[20, 24, 125, 50, 75, 196, 22, 86, 111, 74, 64, 112, 19, 42, 23, 8]<\/p>\n
Number of fano LP in each class is as follows (total 1055):
\n[0, 20, 0, 47, 61, 187, 27, 123, 195, 140, 56, 109, 17, 42, 23, 8]
\nwhich is
\n[0,17+3,0,44+3,61,187,16+11,71+52,107+88,70+70,56,109,17,42,23,8]<\/p>\n
Our enumeration and “fano” values for the extra coefficient are as follows:<\/p>\n
number – associated vertex Laurent polynomial – class – #poly – values [number of appearances]<\/p>\n
0 – x+y+1\/x\/y – P^2 – 20 – nothing
\n1 – y + x\/y + 1\/x\/y – Q – 24 – 0 [17], 4 [3]
\n2 – x+y+xy+1\/x\/y – S_8 – 125 – nothing
\n3 – x+y+1\/x+1\/y – Q – 50 – 0 [44], 4 [3]
\n4 – y+x+x\/y+1\/x\/y – S_7 – 75 – 1 [61]
\n5 – x+y+1\/x+1\/y+xy – S_7 – 196 – 1 [187]
\n6 – y\/x+1\/x\/y+x^2\/y – S_6 – 22 – 2 [16], 3 [11]
\n7 – y+y\/x+1\/x\/y+x\/y – S_6 – 86 – 2 [71], 3 [52]
\n8 – y+x+1\/x+x\/y+1\/x\/y – S_6 – 111 – 2 [107], 3 [88]
\n9 – x+y+x\/y+1\/x+1\/y+y\/x – S_6 – 74 – 2 [70], 3 [70]
\n10- x+y\/x+1\/x\/y+x^2\/y – S_5 – 64 – 3 [56]
\n11- x+y+y\/x+x\/y+1\/x+1\/y+1\/x\/y – S_5 – 112 – 3 [109]
\n12- y+x+1\/x+x^2\/y+1\/x^2\/y – S_4 – 19 – 4 [17]
\n13- y+y\/x+1\/x\/y+x^2\/y – S_4 – 42 – 4 [42]
\n14- xy+y\/x+x\/y+1\/x\/y – S_4 – 23 – 4 [23]
\n15- 1\/x\/y+x^2\/y+y^2\/x – S_3 – 8 – 6 [8]<\/p>\n
Of course everything is extremal and Hodge-Tate, Notations: p.s. – period sequence fano p.s. – one of 105 p.s. of smooth Fano 3-folds good p.s. – p.s. s.t. P_0(D) has integer roots, where L = sum_{i=0}^r t^i P_i(D) is the diff.op. annihilating p.s. bad p.s. – not good p.s. LP – Laurent polynomial. fano LP – Laurent polynomial for which RG coincides […]<\/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-5429","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\/5429","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=5429"}],"version-history":[{"count":1,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts\/5429\/revisions"}],"predecessor-version":[{"id":5430,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts\/5429\/revisions\/5430"}],"wp:attachment":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5429"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5429"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5429"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\nbut some are not-Minkowski, and not even SCR-equivalent (using only surface mutations of the respective facet) to any Minkowski
\n(in particular examples with class Q and coefficient a=4,
\nalso those in Q with coeff a=0 are not lattice Minkowski, but Minkowski).
\n1-puzzles<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"