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.
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 ℤ2-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 |
The total coordinate ring is not yet implemented when the toric variety is degenerate or the class group has torsion.