cartierDivisorGroup X
The group of torus-invariant Cartier divisors on $X$ is the subgroup of all locally principal torus-invariant Weil divisors. On a normal toric variety, the group of torus-invariant Cartier divisors can be computed as an inverse limit. More precisely, if $M$ denotes the lattice of characters on $X$ and the maximal cones in the fan of $X$ are $sigma_0, sigma_1, \dots, sigma_{r-1}$, then we have $CDiv(X) = ker( \oplus_{i} M/M(sigma_i{}) \to{} \oplus_{i<j} M/M(sigma_i \cap sigma_j{})$. For more information, see Theorem 4.2.8 in Cox-Little-Schenck's Toric Varieties.
When $X$ is smooth, every torus-invariant Weil divisor is Cartier.
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On a simplicial toric variety, every torus-invariant Weil divisor is $\QQ$-Cartier; every torus-invariant Weil divisor has a positive integer multiple that is Cartier.
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In general, the Cartier divisors are only a subgroup of the Weil divisors.
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To avoid duplicate computations, the attribute is cached in the normal toric variety.