A syzygetic approach to the smoothability of zero-dimensional schemes

TL;DR

This paper introduces a syzygetic invariant to analyze the smoothability of zero-dimensional schemes, providing new criteria and characterizations for when such schemes deform into collections of distinct points.

Contribution

The paper presents a novel syzygetic invariant that offers obstructions to smoothability and characterizes nonsmoothable schemes of minimal degree across various embedding dimensions.

Findings

Invariant imposes obstructions to smoothability.

Complete characterization of smoothability for certain low-degree schemes.

Identification of nonsmoothable schemes in all embedding dimensions d≥4.

Abstract

We consider the question of which zero-dimensional schemes deform to a collection of distinct points; equivalently, we ask which Artinian k-algebras deform to a product of fields. We introduce a syzygetic invariant which sheds light on this question for zero-dimensional schemes of regularity two. This invariant imposes obstructions for smoothability in general, and it completely answers the question of smoothability for certain zero-dimensional schemes of low degree. The tools of this paper also lead to other results about Hilbert schemes of points, including a characterization of nonsmoothable zero-dimensional schemes of minimal degree in every embedding dimension d\geq 4.