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NormalToricVarieties :: ring(NormalToricVariety)

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

Synopsis

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.

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 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

Caveat

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

See also