2102.00494Jan 31, 202129 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

More elementary components of the Hilbert scheme of points

Mark E. Huibregtse

Algebraic Geometry

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Abstract

Let $K$ be an algebraically closed field of characteristic $0$ , and let $H^μ$ denote the Hilbert scheme of $μ$ points of the affine space $A^n$

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. An elementary component

E

of

H^μ

is an irreducible component such that every

K

-point

[I]

\in

E

represents a length-

μ

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.