Irreducibility of the Gorenstein locus of the punctual Hilbert scheme of degree 10

TL;DR

This paper proves the irreducibility of the Gorenstein locus of the punctual Hilbert scheme for degree 10 in projective space, extending previous results for degrees up to 9, and describes its singular locus.

Contribution

It establishes the irreducibility of the Gorenstein locus for degree 10 and provides a detailed description of its singularities.

Findings

Proves irreducibility of $H_G(10,N)$ for all $N",

Describes the singular locus of the Gorenstein locus in degree 10.

Abstract

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.