INVARIANT HILBERT SCHEMES AND DESINGULARIZATIONS OF QUOTIENTS BY CLASSICAL GROUPS

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 \( \gamma :\mathcal{H}\to W//G \) is a canonical desingularization of the categorical quotient.