A local study of the fiber-full scheme

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.