A theorem of Fogarty states that if $S$ is a smooth algebraic surface, the Hilbert scheme $S^{[n]}$ of length $n$ subschemes of $S$ is smooth for every $n$.

Does anybody know a description of the singularities of $S^{[n]}$ when $S$ is a singular surface? Even $S^{[2]}$ would be enough.

  • $\begingroup$ I think it would be fair to say that no-one knows the answer even for curves. However if you are satisfied to assume $S$ is normal, the situation is much better. $\endgroup$ – Vivek Shende Sep 27 '10 at 16:30
  • $\begingroup$ Can you explain what is known in the normal case? It would already be interesting. $\endgroup$ – Andrea Ferretti Sep 27 '10 at 21:30
  • $\begingroup$ I don't know of a general statement for $S^{[2]}$, but I know that in the simple case of, say, a surface with one node, the singular locus of $S^{[2]}$ is known. I don't remember the details precisely, but I believe it looks locally like $S\times C$, where $C$ is the smooth plane conic $S$ induces (alternatively the exceptional divisor of the blow-up of $S$ at the node). Less geometrically, the singular locus as one might expect consists of length 2 subschemes which are support at least partially at the node. It would probably not be too hard to generalize this to any nodal surface. $\endgroup$ – HNuer Dec 11 '12 at 18:23

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