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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.

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

o2 = true
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.
i3 : W = weightedProjectiveSpace {2,5,7};
i4 : isSimplicial W

o4 = true
i5 : isCartier W_0    

o5 = false
i6 : isQQCartier W_0

o6 = true
i7 : isCartier (35*W_0)      

o7 = true
In general, the Cartier divisors are only a subgroup of the Weil divisors.
i8 : X = normalToricVariety(id_(ZZ^3) | -id_(ZZ^3));
i9 : isCartier X_0

o9 = false
i10 : isQQCartier X_0

o10 = false
i11 : K = toricDivisor X

o11 = - X  - X  - X  - X  - X  - X  - X  - X
         0    1    2    3    4    5    6    7

o11 : ToricDivisor on X
i12 : isCartier K

o12 = true

See also

Ways to use isCartier :