2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/159015Let $k$ be an algebraically closed field and let $\Hilb_{d}^{G}(\p{N})$ be the open locus of the Hilbert scheme $\Hilb_{d}(\p{N})$ corresponding to Gorenstein subschemes. We prove that $\Hilb_{d}^{G}(\p{N})$ is irreducible for $d\le9$, we characterize geometrically its singularities for $d\le 8$ and we give some results about them when $d=9$ which give some evidence to a conjecture on the nature of the singular points in $\Hilb_{d}^{G}(\p{N})$.The exposition has been improved and some of the main results have been extended to degree $d\le 9$Algebraic GeometryCommutative Algebra14C05; 13H10; 14M05text