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 $\mathcal{H}$ 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 $\mathcal{H}$ is a smooth variety, and thus for which the Hilbert-Chow morphism $\gamma: \mathcal{H} \rightarrow W//G$ is a canonical desingularization of the categorical quotient.

Mots clés: Hilbert Function; Smooth Variety; Hilbert Scheme; Borel Subgroup; Vector Bundle

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

Ronan Terpereau. Invariant Hilbert schemes and desingularizations of quotients by classical groups. Transformation Groups, 2014, 19 (1), pp.247-281. ⟨10.1007/s00031-014-9253-1⟩. ⟨hal-00777185v2⟩