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.



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


