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

Synopsis

Usage:

ToricChowRing X
Inputs:
- R, a normal toric variety, A normal toric variety
Outputs:
- a quotient ring,

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 a method function.