4

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.

Michael Hardy
  • 1
user45955
  • 1,055
  • 9
  • 16

1 Answers1

0

A good reference for your question is Ravi Vakil's notes on Foundations of Algebraic Geometry. There is a very nice chapter on Blowups.

To answer your question, I would like to edit your statements a bit. Define $\phi: \mathbb{A}^n \rightarrow \mathbb{A}^s$, by mapping $(x_1, ..., x_n)$ to $(f_1(x), ... ,f_s(x))$, where $x = (x_1, ..., x_n).$ Under $\phi$ the pre-image of $0 \in \mathbb{A}^s$ is precisely $Y$.

We could easily describe the blow-up of $\mathbb{A}^s$ at the origin, $$Bl_0 \mathbb{A}^s = \{ (a, b) \in \mathbb{A}^s \times \mathbb{P}^{s-1}| a_ib_j - a_jb_i = 0 \}.$$

By the Blowup Closure Lemma, the blowup of $\mathbb{A}^n$ at $Y$ is isomorphic to the closure of the image of $\mathbb{A}^n$ in the fiber product $\mathbb{A}^n \times_{\mathbb{A}^s} Bl_0 \mathbb{A}^s$ . The fiber above $Y$ would just be $\mathbb{P}^{s-1}$. Now your question is, how we should describe this blowup in local coordinates. It seems that the coordinates for the blowup $Bl_Y\mathbb{A}^n$ is just $$((x_1, x_2, ..., x_n), \{f_i(x)b_j - b_if_j(x) = 0\}) \subset \mathbb{A}^n \times \mathbb{A}^s \times \mathbb{P}^{s-1}.$$

I am not quite sure whether the equations suggested by you are correct. Note that $n$ may be much bigger than $s$ and your equation just does not make sense. For example, you could try to work out blowup of $\mathbb{A}^{10}$ at $\mathbb{A}^2$ to see what I mean.

Qiao
  • 344
  • 2
  • 10