diff --git a/ToricExtras.m2 b/ToricExtras.m2 index 475abe2..9fb659d 100644 --- a/ToricExtras.m2 +++ b/ToricExtras.m2 @@ -61,7 +61,10 @@ newPackage( { Name => "Will Gilroy", Email => "wlg38@cornell.edu" - } + }, + { + Name => "Rohan Joshi", + Email => "rohansjoshi@math.ucla.edu"} }, Headline => "new routines for working with normal toric varieties", diff --git a/ToricExtras/BatyrevConstructions.m2 b/ToricExtras/BatyrevConstructions.m2 index 9516e82..b5b8a9a 100644 --- a/ToricExtras/BatyrevConstructions.m2 +++ b/ToricExtras/BatyrevConstructions.m2 @@ -90,3 +90,37 @@ batyrevConstructor(List, List, List) := (P, B, C) -> ( ) +end------- + +degreeOfThreefold = method(); +degreeOfThreefold(NormalToricVariety) := X -> ( + K = toricDivisor X; + c = chern(1, OO (-K)); -- anticanonical class in the Chow ring + deg = integral (c*c*c); + -- g = 1 + (deg / 2) + return deg; + + + + +-- There are 7 smooth Fano 3-folds of Picard rank 3. Of these 5 are projectivations of direct sums of two line bundles over surfaces of Picard rank 2, and by Batyrev's paper have 3 primitive collections +-- Therefore, the remaining two (3-26) and (3-29) have Picard rank + +load "BatyrevConstructions.m2" + +X = batyrevConstructor({2,1,1,1,1}, {-2}, {}) +--X = toricProjectiveSpace 3 +K = toricDivisor X +c = chern(1, OO (-K)) +deg = integral (c*c*c) +--g = 1 + (deg / 2) + +X = batyrevConstructor({1,1,1,1,1}, {2}, {}) + +integral (c*c*c) + +isFano X +isSmooth X +isProjective X +isNef(-toricDivisor X) +isAmple(-toricDivisor X) diff --git a/ToricExtras/BatyrevTests.m2 b/ToricExtras/BatyrevTests.m2 index 8a2b7fb..1702834 100644 --- a/ToricExtras/BatyrevTests.m2 +++ b/ToricExtras/BatyrevTests.m2 @@ -1,31 +1,81 @@ TEST /// -- a 2-dimensional fan where the partition sizes of the primitive collections are equal +-- This constructs the unique smooth toric del Pezzo surface of Picard rank 3, which is the Bl_1(P^1 x P^1) = Bl_2(P^2) V = batyrevConstructor({1,1,1,1,1}, {0}, {}) assert(isWellDefined V) assert(isSmooth V) assert(isProjective V) assert(rank picardGroup V == 3) +assert(isFano(V)) /// +TEST /// +-- This constructs a smooth toric weak del Pezzo surface of Picard rank 3, which is the blowup of Bl_1 P^2 at the torus fixed-point on the exceptional divisor +V = batyrevConstructor({1,1,1,1,1}, {1}, {}) +assert(isWellDefined V) +assert(isSmooth V) +assert(isProjective V) +assert(rank picardGroup V == 3) +assert(not isFano(V)) -- not Fano +assert(isNef(-toricDivisor V)) -- but it is weak Fano +/// TEST /// --- a 3-dimensional fan where the partition sizes of the primitive collections are not the same -V = batyrevConstructor({1,1,1,2,1},{0,0},{}) +-- This constructs a smooth toric Fano of Picard rank 3 which is 3-26 on Fanography. It can be described as the blowup of P^3 at the disjoint union of a point and a line +V = batyrevConstructor({2,1,1,1,1}, {0}, {}) assert(isWellDefined V) assert(isSmooth V) assert(isProjective V) assert(rank picardGroup V == 3) +assert(isFano(V)) + +degreeOfThreefold = X -> ( + K := toricDivisor X; + c := chern(1, OO (-K)); + integral (c*c*c) +) + +assert(degreeOfThreefold(V) == 46) + +/// +TEST /// +-- This constructs a smooth toric Fano of Picard rank 3 which is 3-29 on Fanography. It can be described as the blowup of Bl_1 P^3 along a line in the exceptional divisor +V = batyrevConstructor({2,1,1,1,1}, {1}, {}) +assert(isWellDefined V) +assert(isSmooth V) +assert(isProjective V) +assert(rank picardGroup V == 3) +assert(isFano(V)) + +degreeOfThreefold = X -> ( + K := toricDivisor X; + c := chern(1, OO (-K)); + integral (c*c*c) +) + +assert(degreeOfThreefold(V) == 50) + /// TEST /// --- a 12-dimensional fan where the partition sizes of the primitive collection are all different --- 15 rays with 105 maximal cones -V = batyrevConstructor({1,2,3,4,5},{0,0,0,0},{0,0}) +-- a 3-dimensional fan where the partition sizes of the primitive collections are not the same +V = batyrevConstructor({1,1,1,2,1},{0,0},{}) assert(isWellDefined V) assert(isSmooth V) assert(isProjective V) assert(rank picardGroup V == 3) /// +-- GOOD TEST BUT TAKES TOO LONG: +--TEST /// +-- a 12-dimensional fan where the partition sizes of the primitive collection are all different +-- 15 rays with 105 maximal cones +--V = batyrevConstructor({1,2,3,4,5},{0,0,0,0},{0,0}) +--assert(isWellDefined V) +--assert(isSmooth V) +--assert(isProjective V) +--assert(rank picardGroup V == 3) +--/// +