New components of Hilbert schemes of points and 2-step ideals

TL;DR

This paper introduces new examples of irreducible components of Hilbert schemes of points, analyzing deformations of 2-step ideals to reveal reducibility and elementary components in various dimensions.

Contribution

It provides a detailed analysis of 2-step ideals and their deformations, leading to the discovery of numerous new elementary and non-elementary components of Hilbert schemes.

Findings

Identified 215 new generically reduced elementary components in dimensions 4, 5, and 6.

Proved reducibility of certain nested Hilbert schemes and specific cases like $ ext{Hilb}^{3,7} ext{A}^4$.

Developed a method to detect many elementary components via deformation analysis.

Abstract

This paper presents new examples of elementary and non-elementary irreducible components of the Hilbert scheme of points and its nested variants. The results are achieved via a careful analysis of the deformations of a class of finite colength ideals that are introduced in this paper and referred to as 2-step ideals. The most notable reducibility results pertain to the 4-nested Hilbert scheme of points on a smooth surface, the reducibility of $\text{Hilb}^{3,7}\mathbb{A}^4$, and a method to detect a large number of generically reduced elementary components. To demonstrate the feasibility of this approach, we provide an explicit description of 215 new generically reduced elementary components in dimensions 4, 5 and 6.