More elementary components of the Hilbert scheme of points

TL;DR

This paper explores specific irreducible components of the Hilbert scheme of points, providing new examples, simplified methods, and a test indicating many more such components may exist.

Contribution

The authors simplify previous methods and extend their search for elementary components of the Hilbert scheme of points, revealing numerous new examples and proposing a plausibility test for their abundance.

Findings

New examples of elementary components identified

Simplified methods for finding elementary components

A plausibility test suggests many more such components exist

Abstract

Let $K$ be an algebraically closed field of characteristic $0$, and let $H^{\mu}$ denote the Hilbert scheme of $\mu$ points of the affine space $A^n$. An elementary component $E$ of $H^{\mu}$ is an irreducible component such that every $K$-point $[I]$ $\in$ $E$ represents a length-$\mu$ closed subscheme Spec$(K[x_1,\dots,x_n]/I)$ $\subseteq$ $A^n$ that is supported at one point. In a previous article we found some new examples of elementary components; in this article, we simplify the methods and extend the range of the previous paper to find several more examples. In addition, we present a "plausibility test" that suggests the existence of a vast number of similar examples.