From 6721289ab3170a1fec7ef3ccd40f9fc8362915bf Mon Sep 17 00:00:00 2001 From: Rohan Date: Sun, 2 Jun 2024 10:30:13 -0600 Subject: [PATCH] Explicit constructor for surfaces following paper --- ToricExtras/BatyrevSurfaces.m2 | 34 ++++++++++++++++++++++++++++++++++ 1 file changed, 34 insertions(+) create mode 100644 ToricExtras/BatyrevSurfaces.m2 diff --git a/ToricExtras/BatyrevSurfaces.m2 b/ToricExtras/BatyrevSurfaces.m2 new file mode 100644 index 0000000..9d56be5 --- /dev/null +++ b/ToricExtras/BatyrevSurfaces.m2 @@ -0,0 +1,34 @@ + +-- A (normal) projective toric surface of Picard number 3 is specified by five rays in R^2 generated by lattice vectors +-- The surface is smooth if for any two consecutive rays generated by vectors [x_1, y_1], [x_2, y_2], the matrix [[x_1, x_2], [y_1, y_2]] has determinant 1 or -1 +-- Batyrev's paper "On the classification of smooth projective varieties" gives a description of all smooth projective toric varieties (in all dimensions) of Picard number 3 +-- The paper describes such varieties up to isomorphism by giving two sequences of integers {b_i} and {c_i}. Batyrev does not state that all sequences of integers correspond to a *smooth* projective toric variety of Picard rank 3 +-- But there is a variety associated to any sequences {b_i} and {c_i} + +-- In the case of surfaces, there is just one element b and one element c in the sequences +-- Explicitly, the surface associated to b and c is given by the following function + +loadPackage "NormalToricVarieties" + + +rayListGenerator = (b, c) -> {{1,0}, {0,1}, {-1,b+1-c}, {-1,b-c}, {0,-1}} +coneList = {{0,1},{1,2},{2,3},{3,4},{0,4}} +batyrevSurface = (b, c) -> normalToricVariety(rayListGenerator(b, c), coneList) + +-- This is a *particular* solution to the "primitive relations" in Batyrev's paper +-- A convenient feature is the first two rays are [1,0] and [0,1] + +-- It is easy to see that this surface is smooth for any b and c +-- For b = c = 0, we obtain the toric surface Bl_1(P^1 x P^1) (isomorphically, Bl_2 P^2) of Picard number 3 + +-- Further notes: +-- In the case of surfaces, the sets X_0 ... X_4 (in Theorem 6.6) are singleton sets. +-- X_0 = {v}, X_1 = {y}, X_2 = {z}, X_3 = {t}, X_4 = {u} +-- And these go around the fan skipping every other element +-- So our solution is v = [1,0], y = [-1, b+1-c], z = [0, -1], t = [0, 1], u = [1, b-c] +-- These solve the five primitive relations +-- v+y-cz-(b+1)t = 0 +-- y+z-u = 0 +-- z+t = 0 +-- t+u-y = 0 +-- u+v-cz-bt = 0