{"id":134,"date":"2010-06-18T04:00:21","date_gmt":"2010-06-18T04:00:21","guid":{"rendered":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/?p=134"},"modified":"2010-06-19T14:41:03","modified_gmt":"2010-06-19T14:41:03","slug":"unsectionscones-and-tom-vs-jerry-ambiguity-why-no-single-valued-ansatz-is-possible-and-minkowski-ambiguity-is-the-thing-to-expect","status":"publish","type":"post","link":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/?p=134","title":{"rendered":"Unsections\/Cones and “Tom vs Jerry” ambiguity"},"content":{"rendered":"

Unsections\/Cones and “Tom vs Jerry” ambiguity:
\n why no single-valued ansatz is possible and Minkowski ambiguity is the thing to expect<\/strong><\/p>\n

[Miles Reid-like notation]
\nConsider two del Pezzo threefolds of degree 6.
\nLet Jerry be P^1 \\times P^1 \\times P^1
\nand Tom be W = P(T_{P^2}) = X_{1,1} \\subset P^2 \\times P^2 (hyperplane section of product of two planes in Segre embedding).
\nIt is known that Tom and Jerry are not fibers of a flat family.<\/p>\n

Tom has period sequence 6,
\nJerry is grdb[520140 and has period sequence 21.<\/p>\n

Their half-anticanonical section is S = S_6 (del Pezzo surface of degree 6).
\nSo both Tom and Jerry can be degenerated to the same Gorenstein toric Fano threefold — anticanonical cone over S.<\/p>\n

This cone has just one integral point except origin and vertices.
\nLet u = x + y + xy + \\frac{1}{x} + \\frac{1}{y} +\\frac{1}{xy}
\nbe the normalized Laurent polynomial for the honeycomb (fan polytope of S).<\/p>\n

Note that honeycomb has two different Minkowski decompositions — as sum of three intervals and as sum of two triangles.<\/p>\n

These decompositions correspond to two different decompositions of (u+G) into the product of Laurent polynomials [for two different values of G (G=2 and G+3)]:<\/p>\nu+2 = (1 + x) (1 + y) (1 + \\frac{1}{xy})\n

and<\/p>\nu+3 = (1 + x + y) (1 + \\frac{1}{x} + \\frac{1}{y})\n

General Laurent polynomial for the cone over S
\nhas the form<\/p>\nw_G = z (u + G) + \\frac{1}{z}\n

The most interesting thing is the following:<\/p>\n

if we choose $G=2$ then w is mirror of Jerry,
\nbut if we choose $G=3$ then w is mirror of Tom.<\/p>\n

Moreover applying mutation we can transform w to terminal Gorenstein polynomials:<\/p>\n

————
\n[tom]<\/p>\nz (u+2) + \\frac{1}{z} = z (1+x)(1+y)(1+\\frac{1}{xy}) + \\frac{1}{z}\n

becomes z (1+x)(1+y) + \\frac{(1+\\frac{1}{xy})}{z} = z + zx + zy + zxy + \\frac{1}{z} + \\frac{1}{xyz}<\/p>\n

by applying (x,y,z) \\to (x,y,\\frac{z}{1+\\frac{1}{xy}}).<\/p>\n

This corresponds to STD of Tom.
\nIt looks nicer after monomial transformation (x,y,z) \\to (x,y,\\frac{z}{xy}):
\n\\frac{z}{xy} + \\frac{z}{x} + \\frac{z}{y} + \\frac{xy}{z} + \\frac{1}{z}<\/p>\n

————
\n[jerry]<\/p>\nz (u+3) + \\frac{1}{z} = z (1+x+y)(1+\\frac{1}{x}+\\frac{1}{y}) + \\frac{1}{z}\n

becomes z (1+x+y) + \\frac{1 + \\frac{1}{x} + \\frac{1}{y}}{z} = z + zx + zy + \\frac{1}{z} + \\frac{1}{zx} + \\frac{1}{zy}<\/p>\n

by applying (x,y,z) \\to (x,y,\\frac{z}{1+\\frac{1}{x}+\\frac{1}{y}})<\/p>\n

This is simply Laurent polynomial for the smooth model of Jerry: <\/p>\n

x+y+z+\\frac{1}{x} + \\frac{1}{y} + \\frac{1}{z}
\nafter monomial transformation
\n(x,y,z) \\to (xz,yz,z).<\/p>\n

————-<\/p>\n

So Laurent phenomenon distinguishes degenerations of different varieties to the same singular and does not mix them.<\/p>\n","protected":false},"excerpt":{"rendered":"

Unsections\/Cones and “Tom vs Jerry” ambiguity: why no single-valued ansatz is possible and Minkowski ambiguity is the thing to expect [Miles Reid-like notation] Consider two del Pezzo threefolds of degree 6. Let Jerry be and Tom be (hyperplane section of product of two planes in Segre embedding). It is known that Tom and Jerry are […]<\/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":[11,10,9,3],"class_list":["post-134","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-cluster","tag-example","tag-geometry","tag-theory"],"_links":{"self":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts\/134","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=134"}],"version-history":[{"count":14,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts\/134\/revisions"}],"predecessor-version":[{"id":138,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts\/134\/revisions\/138"}],"wp:attachment":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=134"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=134"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=134"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}