Smoothable Gorenstein points via marked schemes and double-generic initial ideals

TL;DR

This paper introduces a new approach using marked schemes and double-generic initial ideals to study smoothable Gorenstein points in Hilbert schemes, providing new results on smoothability and the structure of Hilbert schemes.

Contribution

It presents novel methods for analyzing smoothable Gorenstein points, including criteria for smoothability in specific Hilbert functions and insights into the irreducible components of certain Hilbert schemes.

Findings

Gorenstein points with Hilbert function (1,7,7,1) are smoothable over algebraically closed fields.

Hilb_{16}^7 has at least three irreducible components.

Marked schemes facilitate computation of Zariski tangent spaces in Hilbert schemes.

Abstract

Over an infinite field $K$ with $\mathrm{char}(K)\neq 2,3$, we investigate smoothable Gorenstein $K$-points in a punctual Hilbert scheme from a new point of view, which is based on properties of double-generic initial ideals and of marked schemes. We obtain the following results: (i) points defined by graded Gorenstein $K$-algebras with Hilbert function $(1,7,7,1)$ are smoothable, in the further hypothesis that $K$ is algebraically closed; (ii) the Hilbert scheme $\mathrm{Hilb}_{16}^7$ has at least three irreducible components. The properties of marked schemes give us a simple method to compute the Zariski tangent space to a Hilbert scheme at a given $K$-point, which is very useful in this context. Over an algebraically closed field of characteristic $0$, we also test our tools to find the already known result that points defined by graded Gorenstein $K$-algebras with Hilbert function $(1,5,5,1)$ are smoothable. In characteristic zero, all the results about smoothable points also hold for local Artin Gorenstein $K$-algebras.