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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. 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.

See also