ToricChowRing -- Computes the Chow ring of a normal toric variety

Synopsis

• Usage:
ToricChowRing X
• Inputs:
• R, , A normal toric variety
• Outputs:

Description

Let X be a toric variety with total coordinate ring (Cox ring) R. This method computes the Chow ring Chow ring Ch=R/(SR+LR) of X; here SR is the Stanley-Reisner ideal of the corresponding fan and LR is the ideal of linear relations amount the rays. It is needed for input into the methods Segre, Chern and CSM in the cases where a toric variety is also input to ensure that these methods return results in the same ring. We give an example of the use of this method to work with elements of the Chow ring of a toric variety

 i1 : needsPackage "NormalToricVarieties" o1 = NormalToricVarieties o1 : Package i2 : Rho = {{1,0,0},{0,1,0},{0,0,1},{-1,-1,0},{0,0,-1}} o2 = {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}, {-1, -1, 0}, {0, 0, -1}} o2 : List i3 : Sigma = {{0,1,2},{1,2,3},{0,2,3},{0,1,4},{1,3,4},{0,3,4}} o3 = {{0, 1, 2}, {1, 2, 3}, {0, 2, 3}, {0, 1, 4}, {1, 3, 4}, {0, 3, 4}} o3 : List i4 : X = normalToricVariety(Rho,Sigma,CoefficientRing =>ZZ/32749) o4 = X o4 : NormalToricVariety i5 : R=ring X o5 = R o5 : PolynomialRing i6 : Ch=ToricChowRing(X) o6 = Ch o6 : QuotientRing i7 : describe Ch ZZ[x ..x ] 0 4 o7 = ----------------------------------------- (x x , x x x , x - x , x - x , x - x ) 2 4 0 1 3 0 3 1 3 2 4 i8 : r=gens R o8 = {x , x , x , x , x } 0 1 2 3 4 o8 : List i9 : I=ideal(random({1,0},R)) o9 = ideal(107x + 4376x - 6316x ) 0 1 3 o9 : Ideal of R i10 : K=ideal(random({1,1},R)) o10 = ideal(3187x x - 6053x x - 16090x x + 3783x x + 8570x x + 8444x x ) 0 2 1 2 2 3 0 4 1 4 3 4 o10 : Ideal of R i11 : c=Chern(Ch,X,I) 2 2 o11 = 4x x + 2x + 2x x + x 3 4 3 3 4 3 o11 : Ch i12 : s=Segre(Ch,X,K) 2 2 o12 = 3x x - x - 2x x + x + x 3 4 3 3 4 3 4 o12 : Ch i13 : s-c 2 2 o13 = - x x - 3x - 4x x + x 3 4 3 3 4 4 o13 : Ch i14 : s*c 2 2 o14 = 2x x + x + x x 3 4 3 3 4 o14 : Ch

For the programmer

The object ToricChowRing is .