Relative VGIT and an application to degenerations of Hilbert schemes

TL;DR

This paper extends the semi-continuity theorem for GIT stability to relative settings and applies it to study degenerations of Hilbert schemes, providing a conceptual understanding of stability in these families.

Contribution

It generalizes classical GIT semi-continuity to relative cases and offers new insights into stability of degenerating Hilbert schemes.

Findings

Generalized semi-continuity theorem for GIT in relative settings

Provided a conceptual interpretation of stability loci in degenerating Hilbert schemes

Applied the theory to moduli problems involving degenerations

Abstract

We generalize the classical semi-continuity theorem for GIT (semi)stable loci under variations of linearizations to a relative situation of an equivariant projective morphism from X to an affine base S. As an application to moduli problems, we consider degenerations of Hilbert schemes, and give a conceptual interpretation of the (semi)stable loci of the degeneration families constructed by Gulbrandsen-Halle-Hulek.