Invariant Hilbert Schemes and Luna's Etale Slice Theorem

TL;DR

This paper explores how Luna's etale slice theorem can be applied to analyze the local structure and smoothness of invariant Hilbert schemes, especially at closed orbits, providing new insights into their geometric properties.

Contribution

It demonstrates the novel application of Luna's etale slice theorem to invariant Hilbert schemes, revealing conditions for their smoothness at closed orbits.

Findings

Invariant Hilbert schemes can be studied locally using Luna's slice theorem.

Conditions for smoothness of invariant Hilbert schemes at closed orbits are identified.

The approach offers new tools for analyzing the local geometry of invariant Hilbert schemes.

Abstract

Luna's etale slice theorem is a useful theorem for the local study of quotients by reductive algebraic groups. In this article, we show that the slice theorem can also be used to study local structures of invariant Hilbert schemes. By using this method, we show some results on smoothness of invariant Hilbert schemes at closed orbits.