Embedding the Picard group inside the class group: the case of $\Q$-factorial complete toric varieties

Let $X$ be a $\Q$-factorial complete toric variety over an algebraic closed field of characteristic $0$. There is a canonical injection of the Picard group ${\rm Pic}(X)$ in the group ${\rm Cl}(X)$ of classes of Weil divisors. These two groups are finitely generated abelian groups; whilst the first one is a free group, the second one may have torsion. We investigate algebraic and geometrical conditions under which the image of ${\rm Pic}(X)$ in ${\rm Cl}(X)$ is contained in a free part of the latter group.

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