More elementary components of the Hilbert scheme of points

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.

arXiv comment29 pages, ancillary Mathematica files accompany submission. This revision is in response to referee comments: the construction of modifications is generalized and simplified, leading to more (and more general) examples of elementary components, and a new final section describes a plausibility test that suggests the existence of a large number of additional examples