Irreducibility of the Gorenstein loci of Hilbert schemes via ray families

TL;DR

This paper investigates the structure of the Gorenstein locus in the Hilbert scheme of points on projective space, establishing irreducibility for degrees up to 13 and describing components at degree 14, using ray family techniques.

Contribution

It introduces new criteria for smoothability and smoothness of Gorenstein points, proving irreducibility for degrees up to 13 and characterizing components at degree 14 without computer algebra.

Findings

Gorenstein locus is irreducible for d ≤ 13.

Identifies components of the Gorenstein locus at d=14.

Provides equations for the fourth secant variety of Veronese embeddings.

Abstract

We analyse the Gorenstein locus of the Hilbert scheme of $d$ points on $\mathbb{P}^n$ i.e. the open subscheme parameterising 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 $d\leq 13$ 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 $d\geq 4$.