Irreducibility of the Gorenstein locus of the punctual Hilbert scheme of degree 10

Let $k$ be an algebraically closed field of characteristic 0 and let $H_G(d,N)$ be the open locus of the Hilbert scheme $H(d,N)$ corresponding to Gorenstein subschemes of degree $d$ in the projective N-space. We proved in a previous paper that $H_G(d,N)$ is irreducible for $d\le 9$ and $N\ge 1$. In the present paper we prove that also $H_G(10,N)$ is irreducible for each $N\ge 1$, giving also a complete description of its singular locus.