Invariant Hilbert schemes and desingularizations of quotients by classical groups

TL;DR

This paper studies invariant Hilbert schemes for classical groups acting on classical representations, showing they can provide smooth resolutions of quotient singularities.

Contribution

It constructs examples where invariant Hilbert schemes are smooth and serve as canonical desingularizations of quotient spaces for classical groups.

Findings

Invariant Hilbert schemes can be smooth for classical groups.

These schemes provide canonical desingularizations of quotient singularities.

The Hilbert-Chow morphism acts as a resolution in these cases.

Abstract

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.