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.
Categories: math.AG