= Toric divisor
Every <toric ray> $\rho$ of a <normal variety>[normal] <toric variety> determines a torus-invariant prime <Weil divisor> $D_\rho$. An invariant toric <toric divisor> is an <integer> combination $D=\sum_\rho a_\rho D_\rho$.
Such a <toric divisor> is Cartier exactly when each <toric cone> $\sigma$ admits a <algebraic torus character> $m_\sigma\in M$ satisfying
$$
\langle m_\sigma,v_\rho\rangle=-a_\rho\qquad(\rho\subseteq\sigma).
$$
Then the corresponding local <algebraic torus character> sections trivialize its <divisor line bundle>. Without this integral local condition, the <toric divisor> gives a divisorial rank-one reflexive sheaf and need not define a <line bundle>. The positive class generator of $\mathbb P(1,1,2)$, for example, has Cartier index two. Thus an arbitrary combination of invariant prime <toric divisors> must not be silently identified with invertible-sheaf data.<Toric Varieties by David A. Cox and others>[chapter4, invariant Weil and Cartier divisors]{ref}
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