Relative VGIT and an application to degenerations of Hilbert schemes

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.