Let $u_1=e_1,\quad u_0=e_1+2e_2,\quad u_2=e_2$. Consider the fan consisting of the following cones $\sigma_1= \langle u_1,u_0\rangle$, $\sigma_2=\langle u_0,u_2\rangle$ and their faces. Then the toric variety, $X$, corresponding to this fan is described as follows -
$U_{\sigma_1}=\text{ Maxspec }\mathbb C[x^2y^{-1},y]$, $U_{\sigma_2}=\text{ Maxspec }\mathbb C[x,x^2y]$ are the affine pieces glued using the data given by the natural $\mathbb C$ - algebra isomorphism $g_{2,1}^*:\mathbb C[x^2y^{-1},y]_{x^2y^{-1}}\simeq\mathbb C[x,x^{-2}y]_{x^{-2}y}$.
Nowif $u_0$ were $e_1+e_2$ then I know that the corresponding toric variety is $\mathcal Z(x_0y-x_1x)\subseteq \mathbb {P^1\times C^2}$ which is the blowup of $\mathbb C^2$ at the origin. (Here $(x_0,x_1)$ are the homogeneous coordinates on $\mathbb P^1$ and $(x,y)$ are the coordibates on $\mathbb C^2$)
My question is - can we similarly see $X$ as the blowup of $\mathbb C^2$ at some point? I'm having trouble with beginning this one.
Thank you.