Explicit constructor for surfaces following paper

ToricExtras/BatyrevSurfaces.m2

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+
+-- 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