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 $d\le9$. Moreover we also give a complete picture of its singular locus in the same range $d\le9$. Such a description of the singularities gives some evidence to a conjecture on the nature of the singular points in $\Hilb_{d}^{G}(\p{N})$ that we state at the end of the paper