The fiber-full scheme

Abstract

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 $f:X\subset {\mathbb{P}}_S^r\to S$ such that $R^i{f}_{\ast }\left(\mathcal{G}(\hat{I}½)\right)$ is a locally free ${\mathcal{O}}_S$-module of rank equal to $h_i(\hat{I}½)$, where $\mathbf{h}=(h_0,\dots ,h_r):{\mathbb{Z}}^{r+1}\to {\mathbb{N}}^{r+1}$ 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 Gröbner degenerations.

To appear in Journal of Pure and Applied Algebra