torusIntersection -- compute the ideal of the intersection of a subvariety of a toric variety with the torus

Synopsis

Usage:

torusIntersection(X,I)
Inputs:
- X, a normal toric variety,
- I, an ideal, inside the Cox ring of X
Outputs:
- J, an ideal,

Description

This function calculates the ideal of the intersection of subavariety of a toric variety X with the torus of X. The output is a ideal (saturated with respect to the product of variables) inside the Laurent ring of the torus. The subvariety of X is given by its ideal in the Cox ring. It applies the isomorphism of the Laurent polynomials with the degree zero part of the localization with respect to the product of variables of the Cox ring.

i1 : X = NormalToricVarieties$cartesianProduct apply((1,1), i-> toricProjectiveSpace i);

i2 : S = ring X;

i3 : f = 2 * S_1 + 3 * S_3 + S_0;

i4 : torusIntersection(X,f)

o4 = 2y  + 3y  + 1
       1     2

o4 : QQ[y ..y ]
         1   2

i5 : torusIntersection(X,ideal(f))

o5 = ideal(2y  + 3y  + 1)
             1     2

o5 : Ideal of QQ[y ..y ]
                  1   2

i6 : X = toricProjectiveSpace 3;

i7 : S = ring X;

i8 : f = S_0*S_1*S_2+S_0*S_1*S_3+S_0*S_2*S_3+S_1*S_2*S_3;

i9 : torusIntersection(X,f)

o9 = y y y  + y y  + y y  + y y
      1 2 3    1 2    1 3    2 3

o9 : QQ[y ..y ]
         1   3

i10 : I = ideal((S_1+S_2)*S_0, (S_1+S_2)*(S_0+S_1));

o10 : Ideal of S

i11 : torusIntersection(X,I)

o11 = ideal(y  + y )
             1    2

o11 : Ideal of QQ[y ..y ]
                   1   3

Ways to use torusIntersection:

torusIntersection(NormalToricVariety,Ideal)
torusIntersection(NormalToricVariety,Ideal,Ring)
torusIntersection(NormalToricVariety,RingElement)
torusIntersection(NormalToricVariety,RingElement,Ring)

For the programmer

The object torusIntersection is a method function.