Why do we care about the Hilbert scheme of points?

If $X$ is a scheme, the Hilbert scheme of points $X^{[n]}$ parameterizes zero dimensional subschemes of $X$ of degree $n$.

Why do we care about it?

Of course, there are lots of "in subject" reasons, which I summarize by saying that $X^{[n]}$ is maybe the simplest modern moduli space, and as such is an extremely fertile testing ground for ideas in moduli theory. But it is not clear that this would be very convincing to someone who was not already interested in $X^{[n]}$.

The question I am really asking is:

Why would someone who does not study moduli care about $X^{[n]}$?

The main reason I ask is for the sake of having some relevant motivation sections in talks. But an answer to the following version of the question would be extremely valuable as well:

What can someone who knows a lot something* about $X^{[n]}$ contribute to other areas of algebraic geometry, or mathematics more generally, or even other subjects?

*reworded in light of the answer of Nakajima

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What can someone who knows a lot about $X^{[n]}$ contribute to other areas of algebraic geometry, or mathematics more generally, or even other subjects?

I know a little about $X^{[n]}$. And I have no contribution to mathematics nor other areas of algebraic geometry. But I find study of Hilbert schemes is very interesting. Isn't it enough to motivate to study Hilbert schemes ?

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If I am asked this question by a politician, I will answer differently. –  Hiraku Nakajima Sep 29 '10 at 12:43
-1 mark for excessive modesty –  Richard Thomas Oct 20 '10 at 13:38

One of the reasons is the following. If $X$ is a smooth curve the symmetric product $X^{(n)}$ is smooth. If $X$ is a smooth surface, $X^{(n)}$ is singular, but it is a theorem of Fogarty that $X^{[n]}$ is smooth. So it is a (rather natural) resolution of singularities of $X^{(n)}$.

If moreover $X$ is a symplectic surface (i.e. either a $K3$ or an abelian surface), $X^{[n]}$ has a symplectic form. As Beauville showed, in the $K3$ case it is even an irreducible symplectic variety; in the abelian case a certain subvariety $K_{n-1}(X) \subset X^{[n]}$ is irreducible symplectic.

There are really few known examples of irreducible symplectic varieties in higher dimension: up to deformation, I have listed them all, except for two sporadic examples constructed by O'Grady! This shows that the Hilbert schemes $X^{[n]}$ are indeed very relevant to people who study holomorphic symplectic geometry.

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Andrea- I assume you mean irreducible projective symplectic varieties? There are loads of quasi-projective ones that don't fit in your class. –  Ben Webster Sep 29 '10 at 3:03
@Ben: thank you for pointing out; of course I am talking about complete examples. –  Andrea Ferretti Sep 30 '10 at 19:40

Mark Haiman used Hilbert schemes of surfacesto prove the Macdonald positivity conjecture about Macdonald polynomials: see Hilbert schemes, polygraphs, and the Macdonald positivity conjecture

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The connection between the geometry of the Hilbert scheme of points on $\mathbb{C}^d$ and the combinatorics of $d$-dimensional partitions (piles of $d$-dimensional boxes in the corner of an $d$-dimensional room) may resonate with some audiences. In dimension 3 especially, geometric knowledge has led to beautiful combinatorial results about 3 dimensional box counting (I have in mind Young arXiv:0802.3948 and Okounkov-Reshetikhin-Vafa hep-th/0309208).

The connection that Allen mentions with quivers is nice, and to counter to his dimension 2 bias, I think the quiver story is arguably even better in dimension 3. The Hilbert scheme of points on $\mathbb{C}^3$ (or $\mathbb{C}^3/G$) is given by representations of a quiver with super-potential. Unlike the surface case, the relations on the quiver are given by the critical locus of a single function --- a phenomenon special to dimension 3.

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I believe everyone so far has said "Hilbert schemes of points on surfaces" are worth studying. Personally, I take this as a sign that Hilbert schemes of points (on general $X$) are not that interesting, and that it's merely a historical accident that these glittering jewels were first found within the many tons of worthless Hilbert scheme ore.

In particular, note that the Hilbert scheme of $n$ points in the plane (or on other $\widetilde{{\mathbb C}^2/\Gamma}$, for $\Gamma \leq SU(2)$) can be alternately seen as a Grojnowski-Nakajima quiver variety, and these are smooth holomorphic symplectic manifolds in general.

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Have you heard of any type of process that gives you the Hilbert scheme of points if you input a surface but gives you something better behaved than the Hilbert scheme of points if you plug in something higher-dimensional? –  Peter Samuelson Sep 29 '10 at 5:07

Hilbert schemes of points in the plane come up in representation theory. More specifically in the representation theory of rational Cherednik algebras.

Rational Cherednik algebras, and more generally symplectic reflection algebras have a close connection with symplectic resolutions, algebraic combinatorics, integrable systems and more and are thus very interesting.

See for example: This paper

and This paper

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Slogan: the Hilbert scheme of $n$ points on the minimal resolution of an ADE singularity (associated to a finite subgroup $G \subseteq SL_2(\mathbb{C})$) is to the symplectic reflection algebra of type $G^n \rtimes S_n$ as the flag variety $G/B$ is to the universal enveloping algebra of $\mathrm{Lie}(G)$. So far this is not completely understood. –  Sheikraisinrollbank Sep 28 '10 at 9:43
er... typo. Of course I meant, cotangent bundle to $G/B$. –  Sheikraisinrollbank Sep 29 '10 at 13:42

One can use Hilbert schemes of points on a surface $S$ to detect sigularities of embedded curves $C \subset S$. For instance let $\xi \in S^{[3]}$ be a "fat point", i.e. a subscheme of $S$ isomorphic to $Spec(\mathbb{C}[x,y]/(x^2,xy,y^2))$, then $C$ contains $\xi$ if and only if $C$ has a singularity at $supp(\xi)=\{ p \}$.

In a recent preprint Thomas, Kool and Shende (http://front.math.ucdavis.edu/1010.3211) use this method to proof Goettsche's conjecture about counting singular members of linear systems on surfaces. The basic idea is to write the counting problem as an intersection product of class on $S \times S^{[n]}$ and use results of Ellingsrud-Goettsche-Lehn (building on work of Nakajima) which describe the intersection theory on $S^{[n]}$.

REMARK: I just got aware of the fact, that V. Schende who coauthored the above paper is actually asking the question. So he probably knows about this application.

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+1 for the remark :-) –  Kevin H. Lin Oct 19 '10 at 6:47

Someone answered, when asked by braids and braid groups were important, answered that they are important because they are histories of permutations and permutations are important (or something along that line; I'd love a precise quotation!)

The Hilbert scheme is, in a way, the collection of all possible states in those histories (and all the possible ways you can mess up when permuting points...)

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I would really love any more precise information about these ideas... –  Vivek Shende Sep 27 '10 at 17:48
See the work of Nakajima and Grojnowski, where for a fixed algebraic surface $X,$ the points of the Hilbert schemes $X^{[n]}$ for all $n$ are interpreted as instantons on $X.$ –  Victor Protsak Sep 28 '10 at 3:28

As a student I once asked John Mather for advice about dealing with spaces of ordered $n$-tuples of distinct points in a manifold -- how to get organized about limiting cases where the points come together. He said I needed the Hilbert scheme, and he was right. Well, I wasn't doing algebraic geometry -- I was doing something with smooth manifolds -- and he knew it, so "scheme" was the wrong word. But some kind of essence of the Hilbert scheme idea, adapted to what would have been real semi-algebraic geometry if I had been systematic about it, did the trick.

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If I understand things right, the Hilbert scheme of points in $\mathbb{A}^2$ is hyperkahler manifold and the $n^\textrm{th}$ Calogero-Moser $CM_n$ space can be obtained by deforming the complex structure. The disjoint union of the $CM_n$ parametrizes isomorphism classes of (right) ideals in the first Weyl algebra (polynomial differential operators in 1 variable). This classification is related to Hyugen's principle in differential equations.

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One context in which Hilbert schemes provide a useful conceptual picture is in the Fourier expansion of partition functions that are associated to certain Siegel modular forms of genus 2.

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