On the Gorenstein locus of some punctual Hilbert schemes

Let $k$ 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 $\Hilb_{d}^{G}(\p{N})$.