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 $\{U_i\}$ such that $D|_{U_i}$ is principal in $U_i$ 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 $\QQ$-Cartier, which means that every torus-invariant Weil divisor has a positive integer multiple that is Cartier.
i3 : W = weightedProjectiveSpace {2,5,7};
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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 |