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NormalToricVarieties :: isCartier

isCartier -- 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.

PP3 = projectiveSpace 3;
all(3, i -> isCartier PP3_i)
On a simplicial toric variety, every torus-invariant Weil divisor is -Cartier --- every torus-invariant Weil divisor has a positive integer multiple that is Cartier.
W = weightedProjectiveSpace {2,5,7};
isSimplicial W
isCartier W_0
isQQCartier W_0
isCartier (35*W_0)
In general, the Cartier divisors are only a subgroup of the Weil divisors.
X = normalToricVariety(id_(ZZ^3) | -id_(ZZ^3));
isCartier X_0
isQQCartier X_0
K = toricDivisor X
isCartier K

See also

Ways to use isCartier :