I have just learned how to construct a toric variety from a fan and I am a bit confused. Let $\Sigma$ be the fan that consists of the coordinate axes in $\mathbb{R}^2$, i.e. $\Sigma = \{ \sigma_0, \sigma_1, \sigma_2, \sigma_3, \sigma_4 \}$, where $\sigma_0=\{ (0,0) \}$, $\sigma_1=$Cone$(e_1)$, $\sigma_2=$Cone$(e_2)$, $\sigma_3=$Cone$(-e_1)$, and $\sigma_4=$Cone$(-e_2)$.
What is the toric variety corresponding to this fan?
Denote the coordinate ring of a variety $X$ by $A(X)$ and denote the affine toric variety corresponding to a cone $\sigma$ by $U_{\sigma}$. Is the following true?
- $A(U_{\sigma_1})=\mathbb{C}[y,y^{-1}]$
- $A(U_{\sigma_2})=\mathbb{C}[x,x^{-1}]$
- $A(U_{\sigma_3})=\mathbb{C}[y,y^{-1}]$
- $A(U_{\sigma_4})=\mathbb{C}[x,x^{-1}]$
If so, how do I glue these and what is the corresponding variety? I know that each cone in $\Sigma$ intersects only at $\sigma_0$, so I will be gluing along $(\mathbb{C}^{\ast})^2$. I need help doing this example explicitly. Thanks.