On the Gorenstein locus of the punctual Hilbert scheme of degree 11

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 several previous papers that $\Hilb_{d}^{G}(\p{N})$ is irreducible for $d\le10$ and $N\ge1$, characterizing its singular locus. In the present paper we prove that also $\Hilb_{11}^{G}(\p{N})$ is irreducible for each $N\ge1$. We also give some results about its singular locus.