How to recognize a glueing

Question

This is an exercise in chapter 1 of Fulton's book "Introduction to toric varieties".

Let $\Delta$ be the fan consisting of the cones $\sigma_1=\langle e_1, e_2\rangle$ and $\sigma_2=\langle -e_1,-e_2 \rangle$, where $e_1,e_2$ are the standard basis vectors of $\mathbb{R}^2$.

The problem is to identify the toric variety $X(\Delta)$.

I see that $X(\Delta)$ consists of two open affines, $U_{\sigma_1}=\text{Spec } k[x,y]$ and $U_{\sigma_2}=\text{Spec }k[x^{-1},y^{-1}]$, glued together on $U_{\sigma_1 \cap \sigma_2} = k[x,y,x^{-1},y^{-1}] = (\mathbb{C}^*)^2$ with the glueing defined by $x \mapsto x^{-1}$ and $y \mapsto y^{-1}$.

My initial naïve guess was that this would be $\mathbb{P}^2$ but that would imply that $\mathbb{P}^2$ was the union of two affines, which it's not. Also, maybe it is $\mathbb{C}^2$ blown up at (0,0), but I don't see how to see that either.

So my question is: Are my above thoughts on the problem correct? What is a good strategy to solve this kind of problem? (i.e. how to recognize known varieties from a simple fan). Would it be easier to go about constructing the homogeneous coordinate ring?

Answer 1

You've correctly identified the charts. Later on in section 3.1 of Fulton, the torus orbits are described. In particular, there is one orbit for each cone in $\Delta.$

Let's consider the orbits of this first chart. Since $U_{\sigma_1}\cong\mathbb C^2,$ we know right away that the orbits are $(\mathbb C^*)^2,\{0\}\times\mathbb C^*,\mathbb C^*\times\{0\}$ and $\{(0,0)\}.$ These correspond to the origin, the two rays, and the whole cone $\sigma_1\subseteq N_{\mathbb R}.$ The same thing happens in the other chart, and writing the coordinates suggestively, we have orbits $(\mathbb C^*)^2,\{\infty\}\times\mathbb C^*,\mathbb C^*\times\{\infty\},\{(\infty,\infty)\}.$

Since we glue these two charts along the dense open orbit, putting it all together, we see that our variety is just $(\mathbb P^1\times\mathbb P^1)\setminus(\{(\infty,0)\}\cup\{(0,\infty)\}).$ More directly, we can observe that since $\Delta$ is given by removing two two-dimensional cones from the fan of $\mathbb P^1\times\mathbb P^1$ (shown at the bottom of page 21), the variety $X(\Delta)$ comes from removing the two corresponding torus fixed points, which are exactly $(\infty,0)$ and $(0,\infty)$ as above.