Algebra Seminar: Terpereau -- Invariant Hilbert schemes and resolutions of quotient singularities

Ronan Terpereau (Max Planck, Bonn) 

Friday 11 March, 12-1pm, Place: Carslaw 375 

Invariant Hilbert schemes and resolutions of quotient singularities.  

Let G be a classical group (SL(V), GL(V), O(V),...)  and X be 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 G-stable closed subschemes Z of X such that k[Z] is isomorphic to the
regular representation of G.  In this talk, we will see that H is a smooth variety when
the dimension of V is small, but that H is generally singular.  When H is smooth, the
Hilbert-Chow morphism H -> 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 mention some analogue results in the symplectic setting, that is to say 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, and the Hilbert-Chow
morphism is a resolution of their singularities (sometimes a symplectic one).