Irreducibility of the Gorenstein loci of Hilbert schemes via ray families

Abstract

We analyze the Gorenstein locus of the Hilbert scheme of $d$ points on ${\mathbb{P}}^n$ i.e., the open subscheme parameterizing zero-dimensional Gorenstein subschemes of ${\mathbb{P}}^n$ of degree $d$. 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 $d\le 13$ 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 ${\mathbb{P}}^n$ for $d\ge 4$.