# Blowing up at a subvariety

Let $Y\subseteq\mathbb{A}^n$ be an affine variety with $\mathbb{I}(Y)=(f_{1},\ldots,f_{s}) \subseteq k[x_{1},\ldots,x_{n}]$. Define $\psi:\mathbb{A}^n \to \mathbb{P}^{s-1}$ by $\psi=(f_{1},\ldots,f_{s})$. And let $\Gamma$ be the graph of $\psi$ in $\mathbb{A}^n\times\mathbb{P}^{s-1}$. Then the closure of $\Gamma$ is the blowup of $\mathbb{A}^n$ at $Y$.

Then define $X\subseteq\mathbb{A}^n\times\mathbb{P}^{s-1}$ by $Z(\{y_{i} f_{j} - y_{j} f_{i}\})$.

Claim that $X=$ the closure of $\Gamma$.

One direction is kind of trivial, which is $\bar\Gamma\subseteq X$.

I am stuck in the other direction. I can show that $X - \pi^{-1}(Y)$ is isomorphic to $\mathbb{A}^n - Y$. But I have no idea how to deduce from this isomorphism.

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How do we define the map $\psi$ on $Y$, since by definition the $f_i$ all vanish there? –  rondo9 Dec 11 '12 at 19:25