Is there a relation between the singularities and the divisor class group of a simplicial toric variety

Question

Let $\Delta$ be a simplicial fan in the lattice $N\cong\mathbb Z^n$ with $d$ edges and $\{u_1,\cdots,u_d\}$ are the primitive vectors along the edges. Let $A$ be the divisor class group of the corresponding toric variety $X(\Delta)$. If $\{u_1,\cdots,u_d\}$ spans $N_\mathbb R$ then we have an exact sequence $$ 0\to M\to \bigoplus_{i=1}^d\mathbb ZD_{u_i}\to A\to0$$ where $D_{u_i}$ is the prime divisor corresponding to the edge generated by $u_i$ and $M$ is the dual lattice to $N$.

Now $A$ is a finitey generated abelian group and hence we have $A\cong\mathbb Z^l\times H$ where $H$ is a finite group. Also since $X(\Delta)$ is simplicial we know that it has only finite abelian quotient singularities.

We say that $p\in X$ (an irreducible varoety over $\mathbb C$) is a finite quotient singularity if there is a finite group $G\subseteq\text{GL}(n,\mathbb C)$ and analytic open neighbourhoods $p\in U$ in $X$ and $0\in V$ in $\mathbb C^n/G$ that $U\cong V$ as analtyic varieties and $p\mapsto 0$

Now any $p\in X(\Delta)$ is in some $U_\sigma$ where $\sigma\in\Delta$. let dimension of $\sigma$ be $k$. We have $U_\sigma\cong(\mathbb C^k\times(\mathbb C^*)^{n-k})/G_\sigma$ where $G_\sigma$ is obtained as follows - Let $N'$ be the sublattice of $N$ generated by the primitive vectors along the edges in $\sigma$ and extend it to $N''$ a sublattice of finite index in $N$. Then $G_\sigma=N/N''$

Question 1 - Is there some relation between $H$ and $G_\sigma$ for all $\sigma\in\Delta$? For example is $H$ a subgroup of $G_\sigma$?

So given any point $p\in X(\Delta)$ we have a unique smallest cone $\tau$ (of dimension $r$) such that $p\in U_\tau\cong$ an open subset of $\mathbb C^r/G_\tau$. Now for every $k$ - dimensional $\sigma\ge\tau$ we know that $p\in U_\sigma\cong$ an open subset of $\mathbb C^k/G_\sigma$.

Question 2 - Is there any relation between $G_\tau$ and $G_\sigma$?

Thank you.

Answer 1

Question 2 is easy: Let $N_\sigma$ be the sublattice spanned by the primitive edge vectors of $\sigma$ and let $\overline N_\sigma$ be its saturation: $\overline N_\sigma=\mathbb QN_\sigma\cap N$. Then $G_\sigma=\overline N_\sigma/N_\sigma$. Now if $\sigma$ is a face of $\tau$ then $N_\sigma=\overline N_\sigma\cap N_\tau$ and therefore $$ G_\sigma\subseteq G_\tau. $$

Question 1 is a bit more subtle. I claim that if $\dim\sigma=n$ then $H$ (the torsion in the divisor class group of $X$) is isomorphic to a subgroup of $G_\sigma$ but this embedding is not canonical.

For that let $H_\sigma$ be the divisor class group of $U_\sigma$. Restriction to $U_\sigma$ yields a sequence $$ 0\to\bigoplus_{u_i\not\in\sigma}\mathbb Z D_{u_i}\overset\iota\to A\to H_\sigma\to 0 $$ Here $\iota$ is injective since all relations between the $D_{u_i}$ with $u_i\not\in\sigma$ come from invertible functions on $U_\sigma$. But these are all constant since $\sigma$ is of maximal dimension. This implies that the torsion subgroup of $A$ injects into $H_\sigma$.

The group $H_\sigma$ can also be computed as the cokernel of $$ \mathbb Z^n\cong M\to\bigoplus_{u_i\in\sigma}\mathbb ZD_{u_i}. $$ But the matrix defining this map is the transpose of the map defining the injection $N_\sigma\hookrightarrow N\cong\mathbb Z^n$. The cokernel of the latter is $G_\sigma$. Now observe that an integral $n\times n$-matrix has the same Smith normal form as its transpose. So the cokenrels are isomorphic: $H_\sigma\cong G_\sigma$, but not canonically.