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NormalToricVarieties :: isCartier(ToricDivisor)

isCartier(ToricDivisor) -- whether a torus-invariant Weil divisor is Cartier

Synopsis

Description

A torus-invariant Weil divisor D on a normal toric variety X is Cartier if it is locally principal, meaning that X has an open cover {Ui} such that D|Ui is principal in Ui for every i.

On a smooth variety, every Weil divisor is Cartier.

i1 : PP3 = toricProjectiveSpace 3;
i2 : assert all (3, i -> isCartier PP3_i)

On a simplicial toric variety, every torus-invariant Weil divisor is -Cartier, which means that every torus-invariant Weil divisor has a positive integer multiple that is Cartier.

i3 : W = weightedProjectiveSpace {2,5,7};
i4 : assert isSimplicial W
i5 : assert not isCartier W_0
i6 : assert isQQCartier W_0
i7 : assert isCartier (35*W_0)

In general, the Cartier divisors are only a subgroup of the Weil divisors.

i8 : X = normalToricVariety (id_(ZZ^3) | -id_(ZZ^3));
i9 : assert not isCartier X_0
i10 : assert not isQQCartier X_0
i11 : K = toricDivisor X;

o11 : ToricDivisor on X
i12 : assert isCartier K

See also