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

## Synopsis

• Function: ring
• Usage:
ring X
• Inputs:
• X, ,
• Outputs:
• , 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-invarient 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 or the class group has torsion.