Vakil 9.3 F: Fibers over Blow-Up Map

algebraic-geometryblowup

This in in regards to Vakil 9.3 F. I am trying to compute the fiber over any closed point $p$ of $\mathbb P^1_k$ of the map

$$g: \operatorname{Bl}_{(0, 0)} \mathbb A^2_k \to \mathbb P^1_k$$

The definition we are using for the blowup is

$$\operatorname{Bl}_{(0, 0)} \mathbb A^2_k := \mathbb A^2_k \times_k \mathbb P^1_k$$

modulo the relation $xv = yu$, where $x, y$ are the coordinates of $\mathbb A^2_k$ and $u, v$ are the coordinates of $\mathbb P^1_k$.

By definition, I am trying to compute

$$g^{-1}(p) := \operatorname{Bl}_{(0, 0)} \mathbb A^2_k \times_{\mathbb P^1_k} \operatorname{Spec} \kappa(p)$$

where $\kappa(p)$ is the residue field of $p$. Writing this out on the affine open $u \neq 0$, for example, we get

$$g^{-1}(p) = (\mathbb A^2_k \times_k \mathbb A^{2, v/u}_k) / (xv – yu) \times_{\mathbb A^{2, v/u}_k} k$$

or

$$\operatorname{Spec} k[x, y] \otimes_k k[v/u]/(xv – yu) \otimes k$$

I'd like to say, using the given relation on the four variables, that this is

$$\operatorname{Spec} k[x, y, y/x]$$

but that seems off to me. Where have I gone wrong?

Best Answer

Let's think about what the answer should be before we do the computations. The blowup is the collection of $(p,\ell)\in \Bbb A^2\times\Bbb P^1$ where $p\in[\ell]$, so the fiber over any $\ell\in\Bbb P^1$ should just be the line $\ell$ considered as a subset of $\Bbb A^2$.

To verify this in coordinates, assume that our closed point in $\Bbb P^1$ is $[0:1]$ (we can do this WLOG by applying a change of coordinates). Then the fiber over this point is the same as the fiber of $\operatorname{Spec} k[x,y,u]/(x=yu)\to \operatorname{Spec} k[u]$ over $u=0$, and thus the fiber is the spectrum of $k[x,y,u]/(x-yu)\otimes_{k[u]} k[u]/(u)$. Rewriting $k[x,y,u]/(x-yu)\cong k[y,u]$, we see that $k[y,u]\otimes_{k[u]} k[u]/(u)\cong k[y]$, which is exactly what we expected.

Here are some issues I noticed in your work:

  • in the line where you write "$g^{-1}(p) = (\mathbb A^2_k \times_k \mathbb A^{2, v/u}_k) / (xv - yu) \times_{\mathbb A^{2, v/u}_k} k$", the LHS of your fiber product is of dimension 3, while the blowup is two-dimensional. (This error might just be a typo, though - if you replace the 2 in the second copy of $\Bbb A^2$ with a 1, you're doing alright.)
  • When you write down $\operatorname{Spec} k[x, y] \otimes_k k[v/u]/(xv - yu) \otimes k$, you're not precise about what the second tensor product is over, and it appears you've mis-simplified it. Tensoring with $k$ over $k$ is the identity functor and will preserve dimensions of a ring, but tensoring with $k$ over some other ring can change the dimensions in important ways. Remember that $M\otimes_R R/I\cong M/IM$, so if $k$ is really $k[u]/(u)$ and we're tensoring over $k[u]$, then that matters.

If you want to try and redo your work, I would put more time and effort in to understanding what happens with the tensor product at the end.

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