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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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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 $S^{[n]}$.