Clarification on the definition of a quotient singularity

Question

I am working on the quotient construction of a simplicial toric variety as described in chapter 5 of this book. I have tried the following two examples -

1. The fan $\Delta$ in $\mathbb R^2$ consists of the cone $\sigma=\langle e_1,e_1+2e_2\rangle$ and its faces.

2. The fan $\Delta'$ in $\mathbb R^2$ consists of the edge vectors $v_1=e_1,\ v_2=e_1+2e_2,\ v_3=-e_1+3e_2,\ v_4=-e_1-e_2$ and the cones generated by successive pairs.

For 1 the corresponding toric variety is $\mathbb C^2/G$ where $G=\{(t,t)\in(\mathbb C^*)^2\ | \ t^2=1\}\cong\mathbb Z_2$ and for 2 the corresponding toric variety is $(\mathbb C^4\setminus\mathcal Z)/G$ where $G=\{(t_2t_3^4,t_2,t_3,t_2^2t_3^3)\ |\ t_2,t_3\in\mathbb C^*\}$ and $\mathcal Z=(0\times0\times0\times\mathbb C) \cup (0\times0\times\mathbb C\times0)\cup(0\times\mathbb C\times0\times 0)\cup(\mathbb C\times0\times0\times0)$

Fulton's Introduction to Toric Varieties calls the variety in example 1 a cyclic quotient singularity.

Googling "quotient singularity" gave that it is the quotient of an affine variety $V$ by a finite group $G\subseteq \text{Aut }(V)$ and if $G=\mathbb Z/r$ then it is a cyclic quotient singularity.

The second example however has $G\cong\mathbb{C^*\times C^*}$ which is not finite and $V=\mathbb C^2\setminus\mathcal Z$ is a quasi affine variety. Is there a similar name for such a toric variety $V/G$?

Also, in both examples the fan is simplicial and hence the associated toric variety has singular points. Does "singularity" refer to the fact that the toric variety has singular points?

Thank you.

Answer 1

The connection of toric varieties with quotient singularities is actually quite easy to describe. Let $\Delta\subseteq \mathbb R^n$ be a convex cone whose extremal rays are generated by $v_1,\ldots,v_d\in\mathbb Z^n$. We may assume that the $v_i$ are primitive. Let $X$ be the toric variety attached to $\Delta$. If $d$ is bigger than the dimension of $\Delta$ then $X$ is very singular in the closed orbit (e.g., the divisor class group is infinite). On the other side, if the $v_i$ are linearly independent then the singularieties are very mild, namely abelian quotient singularities.

This can be sees as follows. Assume for simplicity that $d=n$, i.e., that the $v_i$ constitute a basis. Let $\Lambda=\langle v_1,\ldots,v_n\rangle_{\mathbb Z}$. That is a subgroup of $\mathbb Z^n$ of finite index. The torus to be embedded is $T=(\mathbb C^*)^n$ which can also be written as $T=\mathbb C^n/\mathbb Z^n$. Now take the larger torus $\tilde T=\mathbb C^n/\Lambda$. Then there is a surjective homomorphism $\phi:\tilde T\to T$ with kernel $K:=\mathbb Z^n/\Lambda$.

The cone $\Delta$ induces also an embedding $\tilde X$ of $\tilde T$ which this time is isomorphic to $\mathbb C^n$ because the $v_i$ do from a $\mathbb Z$-basis of $\Lambda$. Then $\phi$ extends to a map $\tilde X\to X$ which is just the quotient map by $K$. Thus we realized $X$ as a quotient of the vector space $\tilde X$ by a finite abelian group $K$, i.e. $X\cong\mathbb C^n/K$. Since the $v_i$ were chosen to be primitive one can show that $\mathbb C^n/K$ is singular unless $K=1$.

The group $K$ is easy to compute: form the $n\times n$-matrix $M=[v_1,\ldots,v_n]$ and compute its elementary divisors $d_1,\ldots,d_n$. Then $$K\cong\mathbb Z/d_1\mathbb Z\times\ldots\times\mathbb Z/d_n\mathbb Z.$$

For $n=2$ this is particularly easy: Again because the $v_i$ are primitive we have $d_1=1$. On the other hand $d_1\ldots d_n=\det M$. So $d_2=\det M$ if $n=2$. Using this one quckly sees that $K$ in the four fixed points of your second example is cyclic of order $2$, $5$, $3$, and $1$. So the last fixed point is actually not a singularity.

One final remark: The isomorphism type of $K$ does not suffice to completely determine the isomrophism type of the singularity. For that one also needs to know how $K$ acts on $\mathbb C^n$.

Answer 2

You might find Theorem 11.4.8 in Cox-Little-Schenck relevant: the fan which you've called $\Delta$ is simplicial iff the associated toric variety $X_\Delta$, in your notation, has (at worst) abelian finite quotient singularities. That is, $X_\Delta$ admits an open cover corresponding to the open sets $U_\sigma$ for $\sigma\preceq \Delta$ of the form $\mathbb{A}^n / G$ for $G$ a finite abelian group when $X_\Delta$ has no torus factors (that is, when $X_\Delta$ is not isomorphic to a product of a toric variety of smaller dimension and $\mathbb{C}^*$ to some power), and of the form $(\mathbb{A}^k \times (\mathbb{C}^*)^{n-k})/G$ with $G$ a finite abelian group when $X_\Delta$ does have torus factors.