The locus of points of the Hilbert scheme with bounded regularity

TL;DR

This paper characterizes the locus within the Hilbert scheme corresponding to subschemes with regularity below a fixed bound, describing it explicitly as a locally closed subscheme with equations of bounded degree.

Contribution

It provides an explicit description of the locus of points with bounded regularity in the Hilbert scheme as a locally closed subscheme defined by equations of degree at most +2.

Findings

The locus is an open subscheme of the Hilbert scheme.

It can be embedded as a locally closed subscheme in a Grassmannian.

The defining equations have degree +2 or less.

Abstract

In this paper we consider the Hilbert scheme $Hilb_{p(t)}^n$ 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 $r'$. This locus is an open subscheme of $Hilb_{p(t)}^n$ and, for every $s\geq r'$, we describe it as a locally closed subscheme of the Grasmannian $Gr_{p(s)}^{N(s)}$ given by a set of equations of degree $\leq \mathrm{deg}(p(t))+2$ and linear inequalities in the coordinates of the Pl\"ucker embedding.