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CharacteristicClasses :: ToricChowRing

ToricChowRing -- Computes the Chow ring Ch=R/(SR+LR) of a normal toric variety with coordinate ring R, here SR is the Stanley-Reisner ideal of the corresponding fan and LR is the ideal of linear relations amount the rays

Synopsis

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 , x , x , x ]
                   0   1   2   3   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