Let $\sigma$ be the cone in $\mathbb R^2$ given by $\langle e_1, e_1+2e_2\rangle$. The corresponding affine variety $U_\sigma=\mathcal{Z}(x^2-yz)\subseteq \mathbb C^3$
I am trying to understand the Cox quotient construction given in chapter 5 section 1 of this book. Here he says that a simplicial toric variety corresponding to the fan $\Delta$ can be written as the geometric quotient $(\mathbb C^{\Delta(1)}\setminus Z(\Delta))/G$.
Here, we have an exact sequence $0\to M\to\mathbb Z^{\Delta(1)}\to Cl(X(\Delta))\to0$ where $Cl(X(\Delta))$ is the divisor class group of the toric variety $X(\Delta)$. Applying, $\operatorname{Hom}(\_,\mathbb C^*)$ to this exact sequence we get $0\to G\to\mathbb (C^*)^{\Delta(1)}\to T_N\to 0$. The group $G=\operatorname{Hom}(Cl(X(\Delta),\mathbb C^*)$. The set $Z(\Delta)=\bigcup_C\mathcal{Z}(x_\rho|\rho\in C)$ where the union is taken over all primitive collections in $\Delta(1)$.
I am trying to work it out for the example above. The fan in this case is the cone $\sigma$ and all of its edges. So here as there is no primitive collection, $Z(\Delta)=\emptyset$. $Cl(X(\Delta))=\mathbb Z_2$ and hence $G=\mathbb Z_2$. Thus $U_\sigma=\mathbb C^2/\mathbb Z_2$.
But what I can't understand is what action of $\mathbb Z_2$ on $\mathbb C^2$ gives me the orbit space $\mathcal{Z}(x^2-yz)$. In general I am having trouble understanding how $G$ acts on $\mathbb C^{\Delta(1)}\setminus Z(\Delta)$.
I understand that there is a natural action of $(\mathbb C^*)^{\Delta(1)}$ on $(\mathbb C)^{\Delta(1)}$ and $G$ sits inside $(\mathbb C^*)^{\Delta(1)}$ and hence acts on $(\mathbb C)^{\Delta(1)}$. However I don't get how $G$ sits inside $(\mathbb C^*)^{\Delta(1)}$.
Can someone help me out?
Thanks.