What Toric Variety does this fan correspond to?

Question

Let $\Sigma$ be the fan defined by $\{\sigma_1,\sigma_2,\sigma_3,\sigma_4,\star\}$, where $\sigma_1=\operatorname{cone}(e_1)$, $\sigma_2=\operatorname{cone}(e_2)$, $\sigma_3=-\sigma_1$, $\sigma_4=-\sigma_2$, and $\star=\operatorname{cone}((0,0))$. We are working on a two dimensional lattice $N$ extended to the real vector space $N_{\mathbb R}\cong \mathbb R^2$, and $e_i$ are the standard basis vectors.

Now my question is, how do I recognize what toric variety this fan corresponds to? Importantly, I can't quite figure out how things should glue.

Here's my process, since the dual cones are given by: $$\sigma_i^*=\{m\in M_{\mathbb R}:m(u)\geq 0,\forall u\in \sigma_i\}$$ where $M$ is dual lattice, we have that: \begin{alignat}{3} \sigma_1^*=&\operatorname{cone}(-e^2,e^2,e^1)\qquad&\sigma_2^*=&\operatorname{cone}(-e^1,e^1,e^2)\\ \sigma_3^*=&\operatorname{cone}(-e^2,e^2,-e^1)\qquad&\sigma_4^*=&\operatorname{cone}(-e^1,e^1,-e^2) \end{alignat} So we have that: \begin{alignat}{3} U_{\sigma_1}=&\operatorname{Spec}\mathbb C[x,y,y^{-1}]\qquad&U_{\sigma_2}=&\operatorname{Spec}\mathbb C[x,x^{-1},y]\\ U_{\sigma_3}=&\operatorname{Spec}\mathbb C[x^{-1},y,y^{-1}]\qquad&U_{\sigma_4}=&\operatorname{Spec}\mathbb C[x,x^{-1},y^{-1}] \end{alignat} and then all of these glue along the torus in each $U_{\sigma_i}$, but how do I actually deduce the gluing maps? For example take $U_x\subset U_{\sigma_1}$, where $U_x\cong \operatorname{Spec}\mathbb C[x,x^{-1},y,y^{-1}]$, and $U_{x^{-1}}\subset U_{\sigma_3} $ where $U_{x^{-1}}\cong \operatorname{Spec}\mathbb C[x,x^{-1},y,y^{-1}]$. Any isomorphism between these should be an automorphism of ring $\mathbb C[x,x^{-1},y,y^{-1}]$, but how am I supposed to tell what automorphism that should be? Should I just map $x\mapsto x$, and $y\mapsto y$ for each variable? What about the other pieces? I get that the intersections tell me what distinguished opens to glue along, but how do I determine how they glue?

Answer 1

Although this might not be exactly what you want, I think it's helpful to think geometrically and identify $\operatorname{Spec}\mathbb{C}[X^{-1}]$ with $\mathbb{C}^*\cup\infty$. This way, you don't really need to think about the gluing maps, and you can just identify spaces.

In your example, $U_{\sigma_1} = \mathbb{C}\times\mathbb{C}^*$, $U_{\sigma_2} = \mathbb{C}^*\times\mathbb{C}$, $U_{\sigma_3} = (\mathbb{C}^*\cup\infty)\times\mathbb{C}^*$, and $U_{\sigma_4} = \mathbb{C}^* \times(\mathbb{C}^*\cup\infty)$. Notice that $\mathbb{C} = \mathbb{C}^* \cup 0$, and the space that's being identified is the 2-torus $\mathbb{C}^*\times\mathbb{C}^*$. Also, $\mathbb{P}^1 = \mathbb{C} \cup \infty$. Using these facts, you'll be able to find out that the toric variety is $\mathbb{P}^1\times\mathbb{P}^1 - (0\cup\infty)\times(0\cup\infty)$ after some tinkering.

Answer 2

The variety is P1 times P1 minus the four fixed points. Ripping out the big cones in the fan corresponds to vertices in the polytopal picture.