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.