4
$\begingroup$

On a summer school for undergraduate and graduate students Okounkov gave the following exercise (without hints): Prove that the Hilbert scheme of points on $\mathbb C^2$ is smooth.

Only a definition of the scheme was given. I don't really understand how to solve this exercise.

I remember Beauville was saying in his course on a school at Lac de Garde (while he was commenting page 14 here: http://math1.unice.fr/~beauvill/conf/lacgarde2.pdf), that he is not aware of an elementary proof of the fact that the Hilbert scheme of points on a smooth surface is smooth (a standard proof uses $Ext$ groups).

So I would like to know, who is right, Okounkov or Beauville? (or maybe both?...)

Improve this question
$\endgroup$
6
  • $\begingroup$ Do you know about $Ext$ groups? $\endgroup$
    – Will Sawin
    Commented Dec 11, 2013 at 23:37
  • $\begingroup$ Depends on what is meant by "know". I red the definition 20 times over the span of my life. I can calculate these groups in some simple examples. But since I still have not read Hartchorne beyond section 1, maybe the honest answer should be: NO. $\endgroup$
    – aglearner
    Commented Dec 11, 2013 at 23:46
  • $\begingroup$ There are many proofs of this result--one using Ext groups is in FGA Explained, in the chapter on Hilbert schemes of points. It's not actually that bad, in my opinion. I believe one can also prove this result (more elementarily) using the description of the Hilbert scheme as a space of commuting matrices. $\endgroup$
    – Daniel Litt
    Commented Dec 12, 2013 at 2:07
  • 3
    $\begingroup$ I think there's a proof that goes like this. Let $T = (\mathbb C^\times)^2$ act on the plane hence on the Hilbert scheme. Then any point has a $T$-fixed point in the closure of its $T$-orbit, hence it's enough to check smoothness at the $T$-fixed points. For each of those, you can figure out what $Hom_R(I,R/I)$ looks like as a $T$-representation, and in particular that it is $2n$-dimensional. Which is to say, no $Ext^{i>0}$ is considered. $\endgroup$
    – Allen Knutson
    Commented Dec 12, 2013 at 7:38
  • 1
    $\begingroup$ The proof Allen sketches is written out in detail in chapter 18.2 of Miller and Sturmfels <i>Combinatorial Commutative Algebra</i>. It's elementary, but it's not easy. $\endgroup$
    – David E Speyer
    Commented Dec 12, 2013 at 15:26

1 Answer 1

Reset to default
2
$\begingroup$

A way to get the smoothness fro free is by using the description of the Hilbert scheme as a quiver variety. Since it is a result of the hyperkahler reduction applied to a flat variety and the parameter of the reduction is regular (in this case regularity is just its nonvanishing), it is smooth.

Improve this answer
$\endgroup$
2
  • $\begingroup$ Sasha, could you give me some good reference for this? $\endgroup$
    – aglearner
    Commented Dec 12, 2013 at 11:11
  • $\begingroup$ I think the best reference is the Nakajima's book. $\endgroup$
    – Sasha
    Commented Dec 12, 2013 at 19:15

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .