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. Let $f:X\subset P_S^r\to S$ be a projective morphism and $h=(h_0,\dots ,h_r):Z^{r+1}\to N^{r+1}$ be a fixed tuple of functions. The fiber-full scheme ${\mathrm{Fib}}_{F/X/S}^h$ is a fine moduli space parametrizing all quotients $G$ of a fixed coherent sheaf $F$ on X such that $R^i{f}_{⁎}\left(G\right(\nu \left)\right)$ is a locally free $O_S$-module of rank equal to $h_i\left(\nu \right)$. 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 Gröbner degenerations.