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

TL;DR

This paper studies the relationship between the Picard group and the class group of $Q$-factorial complete toric varieties, identifying conditions for the Picard group to embed into a free part of the class group.

Contribution

It provides algebraic and geometric criteria for when the Picard group injects into a free component of the class group in these varieties.

Findings

Conditions for Picard group embedding into free part of class group

Characterization of torsion in the class group

Insights into divisor class structures in toric varieties

Abstract

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.