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 -

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

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.

smoothvariety by a finite group action. $\endgroup$