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