Invariant Hilbert Schemes and Luna's Etale Slice Theorem

TL;DR

The paper uses Luna's étale slice theorem to study local structures of invariant Hilbert schemes ${\rm Hilb}^{G}_{h}(X)$, showing that near a closed orbit the scheme is étale-locally governed by a finite-type data on a slice $V$ and its tangent space $T_xV$ via ${{\rm Hilb}^{G_x}_{h'}}(V)$ and ${{\rm Hilb}^{G_x}_{h'}}(T_xV)$. It proves the existence of finitely many etale morphisms whose images cover open parts of ${\rm Hilb}^{G}_{h}(X)$, relates the invariant Hilbert scheme to the refined quotient when $h(0)=1$, and provides smoothness criteria at points corresponding to closed orbits in terms of the punctual Hilbert schemes of the slice. It also establishes that for a closed orbit, the associated Hilbert function $h'$ has finite support and yields an open subset ${{\rm Hilb}^{G_x}_{h'}}(T_xV)$ inside the punctual Hilbert scheme ${\rm Hilb}_{n}(T_xV)$ with $n=\sum h'(M)$. The paper culminates with a concrete example where $G={\mathbb G}_m^{2}$ acts on ${\mathbb A}^{4}$, illustrating smoothness and providing explicit equations and families for the universal invariant subschemes. This approach offers a practical, local-to-global method for assessing smoothness and structure of invariant Hilbert schemes via slice data.

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.

Theorems & Definitions (21)

• Proposition 2.1: Luna's étale slice theorem
• Proposition 2.2
• Theorem 2.3
• Theorem 2.4
• Corollary 2.5
• Corollary 2.6
• proof
• Definition 3.1
• Lemma 3.2
• proof