MORE ELEMENTARY COMPONENTS OF THE HILBERT SCHEME OF POINTS

Abstract

Let $K$ be an algebraically closed field of characteristic $0$, and let $H_{{\mathbb{𝔸}}_k^n}^{\mu }$ denote the Hilbert scheme of $\mu$ points of the affine space ${\mathbb{𝔸}}_K^n$. An elementary component $E$ of $H_{{\mathbb{𝔸}}_k^n}^{\mu }$ is an irreducible component such that every $K$-point $[I]$ $\in$ $E$ represents a length-$\mu$ closed subscheme $\text{Spec}(k[x_1,\dots ,x_n]∕I)$ $\subseteq$ ${\mathbb{𝔸}}_K^n$ 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.