  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.
