  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 $d\le9$ and $N\ge1$. In the
present paper we prove that also $H_G(10,N)$ is irreducible for each $N\ge1$,
giving also a complete description of its singular locus.
