On the Gorenstein locus of some punctual Hilbert schemes

TL;DR

This paper investigates the structure and singularities of the Gorenstein locus within punctual Hilbert schemes, proving irreducibility for small degrees and providing geometric characterizations of singularities.

Contribution

It establishes the irreducibility of the Gorenstein locus for degrees up to 9 and characterizes its singularities for degrees up to 8, offering evidence for a conjecture on singular points.

Findings

Proved irreducibility of Hilb_{d}^{G}(P^N) for d a9 9

Characterized singularities of Hilb_{d}^{G}(P^N) for d a8 8

Provided evidence supporting a conjecture on the nature of singular points in the Gorenstein locus

Abstract

Let $k$ be an algebraically closed field and let $\Hilb_{d}^{G}(\p{N})$ be the open locus of the Hilbert scheme $\Hilb_{d}(\p{N})$ corresponding to Gorenstein subschemes. We prove that $\Hilb_{d}^{G}(\p{N})$ is irreducible for $d\le9$, we characterize geometrically its singularities for $d\le 8$ and we give some results about them when $d=9$ which give some evidence to a conjecture on the nature of the singular points in $\Hilb_{d}^{G}(\p{N})$.