TITLE: Why do we care about the Hilbert scheme of points?
QUESTION [32 upvotes]: 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

REPLY [8 votes]: 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.