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