Irreducibility of the Gorenstein loci of Hilbert schemes via ray families

We analyze the Gorenstein locus of the Hilbert scheme of $d$ points on $p n$ i.e., the open subscheme parameterizing zero--dimensional Gorenstein subschemes of $p n$ of degree $d$ d. We give new sufficient criteria for smoothability and smoothness of points of the Gorenstein locus. In particular we prove that this locus is irreducible when $dle 13$ and find its components when $d=14$. The proof is relatively self-contained and it does not rely on a computer algebra system. As a by--product, we give equations of the fourth secant variety to the $d$--th Veronese reembedding of $p n$ for $dge4$.