On the irriducibility and singularities of the Gorenstein locus of the punctual Hilbert scheme of degree 10

Let $k$ be an algebraically closed field of characteristic $0$ and let $\Hilb_{d}^{G}(\p{N})$ be the open locus of the Hilbert scheme $\Hilb_{d}(\p{N})$ corresponding to Gorenstein subschemes. We proved in a previous paper that $\Hilb_{d}^{G}(\p{N})$ is irreducible for $d\le9$ and $N\ge1$. In the present paper we prove that also $\Hilb_{10}^{G}(\p{N})$ is irreducible for each $N\ge1$, giving also a complete description of its singular locus.