The fiber-full scheme

We introduce the fiber-full scheme which can be seen as the parameter space that generalizes the Hilbert and Quot schemes by controlling the entire cohomological data. The fiber-full scheme $\text{Fib}_{\mathcal{F}/X/S}^\mathbf{h}$ is a fine moduli space parametrizing all quotients $\mathcal{G}$ of a fixed coherent sheaf $\mathcal{F}$ on a projective morphism $f:X \subset \mathbb{P}_S^r \rightarrow S$ such that $R^i{{f}_*}\left(\mathcal{G}(\nu)\right)$ is a locally free $\mathcal{O}_S$-module of rank equal to $h_i(\nu)$, where $\mathbf{h} = (h_0,\ldots,h_r) : \mathbb{Z}^{r+1} \rightarrow \mathbb{N}^{r+1}$ is a fixed tuple of functions. In other words, the fiber-full scheme controls the dimension of all cohomologies of all possible twistings, instead of just the Hilbert polynomial. We show that the fiber-full scheme is a quasi-projective $S$-scheme and a locally closed subscheme of its corresponding Quot scheme. In the context of applications, we demonstrate that the fiber-full scheme provides the natural parameter space for arithmetically Cohen-Macaulay and arithmetically Gorenstein schemes with fixed cohomological data, and for square-free Gr\"obner degenerations.