A local study of the fiber-full scheme

TL;DR

This paper investigates the local properties of the fiber-full scheme, a moduli space generalizing the Hilbert scheme, and provides conditions for cohomology stability and a tangent-obstruction theory.

Contribution

It introduces new local property analyses of the fiber-full scheme, including cohomology invariance conditions and a tangent-obstruction framework.

Findings

Cohomology remains constant under certain Gr"obner degenerations.

A tangent-obstruction theory analogous to the Hilbert scheme is developed.

Conditions for cohomology stability are established.

Abstract

We study some of the local properties of the fiber-full scheme, which is a fine moduli space that generalizes the Hilbert scheme by parametrizing closed subschemes with prescribed cohomological data. As a consequence, we provide sufficient conditions for cohomology to remain constant under Gr\"obner degenerations. We also describe a tangent-obstruction theory for the fiber-full scheme in analogy with the one for the Hilbert scheme.