On the Gorenstein locus of some punctual Hilbert schemes

Let $k$ be an algebraically closed field and let ${\text{Hilb}}_d^G(\text{p}N)$ be the open locus of the Hilbert scheme ${\text{Hilb}}_d(\text{p}N)$ corresponding to Gorenstein subschemes. We prove that ${\text{Hilb}}_d^G(\text{p}N)$ is irreducible for $d\le 9$, 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 ${\text{Hilb}}_d^G(\text{p}N)$.