The locus of points of the Hilbert scheme with bounded regularity

Abstract

In this paper we consider the Hilbert scheme $Hil{b}_{p(t)}^n$ parameterizing subschemes of $P^n$ with Hilbert polynomial $p(t)$, and we investigate its locus containing points corresponding to schemes with regularity lower than or equal to a fixed integer $r^{\prime }$. This locus is an open subscheme of $Hil{b}_{p(t)}^n$ and, for every $s\ge r^{\prime }$, we describe it as a locally closed subscheme of the Grasmannian $G{r}_{p(s)}^{N(s)}$ given by a set of equations of degree $\le \mathrm{deg}(p(t))+2$ and linear inequalities in the coordinates of the Plücker embedding.

v2: new proofs relying on the functorial definition of the Hilbert scheme. v3: Sections reorganized, new self-contained proof of the representability of the Hilbert functor with bounded regularity (Section 6)

References in corpus (1)

• Flat families by strongly stable ideals and a generalization of Groebner bases

Cited by in corpus (2)

• Double-Generic Initial Ideal and Hilbert Scheme
• Minimal Castelnuovo-Mumford regularity for a given Hilbert polynomial