Invariant Hilbert schemes and resolutions of quotient singularities
Ronan Terpereau (Grenoble)
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When 
Jun 20, 2013 from 03:15 pm to 04:15 pm 
Where  Mainz, 05432 (Hilbertraum) 
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Abstract:
Let G be a classical group (SL_n, GL_n, O_n,...) and X the direct sum of p copies of the standard representation of G and q copies of its dual representation, where p and q are positive integers. We consider the invariant Hilbert scheme, denoted H, which parametrizes the Gstable closed subschemes Z of X such that the coordinate ring k[Z] is isomorphic to the regular representation of G.
In this talk, we will see that H is a smooth variety when n is small, but that H is generally singular. When H is smooth, the HilbertChow morphism H \rightarrow X//G is a canonical resolution of the singularities of the categorical quotient X//G (=Spec(k[X]^G)). Then it is natural to ask what are the good geometric properties of this resolution (for instance if it is crepant).
To finish, we will see some analogue results in the symplectic setting, that is, by letting p=q and replacing X by the zero fiber of the moment map. The quotients that we get by doing this are isomorphic to the closures of nilpotent orbits in simple Lie algebras, and the HilbertChow morphism is a resolution of their singularities (sometimes a symplectic one).