ring(NormalToricVariety) -- make the total coordinate ring (a.k.a. Cox ring)

Synopsis

Function: ring
Usage:

ring X
Inputs:
- X, a normal toric variety,
Outputs:
- a polynomial ring, the total coordinate ring

Description

The total coordinate ring, which is also known as the Cox ring, of a normal toric variety is a polynomial ring in which the variables correspond to the rays in the fan. The map from the group of torus-invariant Weil divisors to the class group endows this ring with a grading by the class group. For more information, see Subsection 5.2 in Cox-Little-Schenck's Toric Varieties.

The total coordinate ring for projective space is the standard graded polynomial ring.

i1 : PP3 = toricProjectiveSpace 3;

i2 : S = ring PP3;

i3 : assert (isPolynomialRing S and isCommutative S)

i4 : gens S

o4 = {x , x , x , x }
       0   1   2   3

o4 : List

i5 : degrees S

o5 = {{1}, {1}, {1}, {1}}

o5 : List

i6 : assert (numgens S == #rays PP3)

i7 : coefficientRing S

o7 = QQ

o7 : Ring

For a product of projective spaces, the total coordinate ring has a bigrading.

i8 : X = toricProjectiveSpace(2) ** toricProjectiveSpace(3);

i9 : gens ring X

o9 = {x , x , x , x , x , x , x }
       0   1   2   3   4   5   6

o9 : List

i10 : degrees ring X

o10 = {{1, 0}, {1, 0}, {1, 0}, {0, 1}, {0, 1}, {0, 1}, {0, 1}}

o10 : List

A Hirzebruch surface also has a $\ZZ^2$-grading.

i11 : FF3 = hirzebruchSurface 3;

i12 : gens ring FF3

o12 = {x , x , x , x }
        0   1   2   3

o12 : List

i13 : degrees ring FF3

o13 = {{1, 0}, {-3, 1}, {1, 0}, {0, 1}}

o13 : List

To avoid duplicate computations, the attribute is cached in the normal toric variety. The variety is also cached in the ring.

Caveat

The total coordinate ring is not yet implemented when the toric variety is degenerate, and is experimental when the class group has torsion.

Ways to use this method: