Local finite dimensional Gorenstein k-algebras having Hilbert function (1,5,5,1) are smoothable

TL;DR

This paper proves that Gorenstein algebras with Hilbert function (1,5,5,1) are smoothable, contributing to the understanding of the structure of the Gorenstein locus in the Hilbert scheme for length 12 algebras.

Contribution

It establishes the smoothability of Gorenstein algebras with Hilbert function (1,5,5,1), a case previously difficult to analyze with non-direct methods.

Findings

Gorenstein algebras with Hilbert function (1,5,5,1) are smoothable.

The result advances the understanding of the Gorenstein locus in Hilbert schemes.

Addresses a challenging case among length 12 Gorenstein algebras.

Abstract

Let k be an algebraically closed field of characteristic 0. The question of irreducibility of the punctual Hilbert scheme Hilb_d P^n and its Gorenstein locus for various d was studied in [CEVV8, CN9, CN10, CN11]. In this short paper we prove that the subschemes corresponding to the Gorenstein algebras having Hilbert function (1,5,5,1) are smoothable i.e. lie in the closure of the locus of smooth subschemes. Among the Gorenstein algebras of length 12 the smoothability of algebras having such Hilbert function seems to be the most inapproachable using non-direct tools e.g. structural theorems.