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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.

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    $\begingroup$ No. The blowup of a point in $\mathbb C^2$ is a smooth variety, but the cone $\sigma_1 = \langle u_0,u_1 \rangle$ is not a smooth cone, so your variety has a singular point. $\endgroup$
    – Fredrik Meyer
    May 2, 2016 at 6:32
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    $\begingroup$ This is a weighetd blowup. Equivalently, this is the blowup of the ideal $(x^2,y)$. $\endgroup$
    – Sasha
    May 2, 2016 at 6:40
  • $\begingroup$ @Sasha, could you explain what you mean by weighted blowup? Does it mean that the blow up is of a non-homogeneous ideal? $\endgroup$
    – R_D
    May 2, 2016 at 6:59
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    $\begingroup$ @Rise: it means that you put a different grading on the algebra $ \oplus I^k$ before taking its Proj. This is like a difference between a projective space and a weighted projective space --- both a projective spectra of the same (polynomial) algebra, but with a different grading. $\endgroup$
    – Sasha
    May 4, 2016 at 8:42

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