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 ℚ-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 |