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Hilbert schemes of orbifold surfaces and counting problems

Algebra & Discrete Mathematics

Speaker: Jason Starr, Stony Brook University
Location: 1147 MSB
Start time: Thu, Jul 3 2008, 3:10PM

The Hilbert scheme of n points on a smooth, projective surface is a compactification of the space of unordered n-tuples of distinct points in the surface. It connects to combinatorics via its Betti and Hodge numbers. An orbifold surface is a projective surface with finitely many orbifold points. The Hilbert schemes of these orbifolds are again smooth and projective. When the surface has ADE singularities, the Betti and Hodge numbers equal those of the Hilbert scheme of the minimal desingularization, and the "local" ring structure on cohomology connects to the McKay correspondence (although the global ring structure appears to be open). But when the surface has non-ADE singularities, even such numbers as the Euler characteristic of the local model, Hilb^n([C^2/G]), appear to be unknown. This comes down to counting Young diagrams with certain patterns. I will report what Li Li and I have learned about these problems, and mostly appeal to the combinatorial expertise of the audience (who I am certain can make more progress than we have).