Torsionfreeness for divisor class groups of toric rings of integral polytopes

Abstract

In the present paper, we give some sufficient conditions for $\mathrm{Cl}(\mathbb{k}[P])$ to be torsionfree, where $\mathrm{Cl}(\mathbb{k}[P])$ denote the divisor class group of the toric ring $\mathbb{k}[P]$ of an integral polytope $P$. We prove that $\mathrm{Cl}(\mathbb{k}[P])$ is torsionfree if $P$ is compressed, and $\mathrm{Cl}(\mathbb{k}[P])$ is torsionfree if $P$ is a $(0,1)$-polytope which has at most $\dim P+2$ facets. Moreover, we characterize the toric rings of $(0,1)$-polytopes in the case $\mathrm{Cl}(\mathbb{k}[P])\cong \mathbb{Z}$.

8 pages, 3 figures