A relative Hilbert-Mumford criterion

TL;DR

This paper extends the Hilbert-Mumford criterion for GIT stability to a relative setting involving equivariant morphisms over noetherian rings, and applies it to moduli problems like degenerations of Hilbert schemes.

Contribution

It generalizes classical GIT stability criteria to relative contexts and proves projectivity of associated GIT quotients in this setting.

Findings

Generalized Hilbert-Mumford criterion for relative GIT stability.

Proved projectivity of the quotient morphism in the relative setting.

Applied results to degenerations of Hilbert schemes of points.

Abstract

We generalize the classical Hilbert-Mumford criteria for GIT (semi-)stability in terms of one parameter subgroups of a linearly reductive group G over a field k, to the relative situation of an equivariant, projective morphism X -> Spec A to a noetherian k-algebra A. We also extend the classical projectivity result for GIT quotients: the induced morphism X^ss/G -> Spec A^G is projective. As an example of applications to moduli problems, we consider degenerations of Hilbert schemes of points.