  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.
