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
ℚ-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 |