On the irreducibility and the 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 be the open locus of the Hilbert scheme corresponding to Gorenstein subschemes. We proved in a previous paper that is irreducible for d≤9 and N≥1. In the present paper we prove that is irreducible for each N≥1, giving also a complete description of its singular locus.