Invariant Hilbert schemes and desingularizations of quotients by classical groups

Ronan Terpereau (1)

1 IF - Institut Fourier

Let W be a finite-dimensional representation of a reductive algebraic group G. The invariant Hilbert scheme H is a moduli space that classifies the G-stable closed subschemes Z of W such that the affine algebra k[Z] is the direct sum of simple G-modules with prescribed multiplicities. In this article, we consider the case where G is a classical group acting on a classical representation W and k[Z] is isomorphic to the regular representation of G as a G-module. We obtain families of examples where H is a smooth variety, and thus for which the Hilbert-Chow morphism \gamma: H \rightarrow W//G is a canonical desingularization of the categorical quotient.

Mots clés: Invariant Hilbert schemes; desingularization; invariant theory; determinantal variety; homogeneous vector bundle.; homogeneous vector bundle

Domaines: Géométrie algébrique [math.AG]; Théorie des représentations [math.RT]

Ronan Terpereau. Invariant Hilbert schemes and desingularizations of quotients by classical groups. 2013. ⟨hal-00777185v1⟩