Some elementary components of the Hilbert scheme of points

TL;DR

This paper explores elementary components of the Hilbert scheme of points in affine space, generalizing previous examples to include new cases beyond homogeneous ideals, enhancing understanding of zero-dimensional subschemes.

Contribution

It extends the known constructions of elementary components of the Hilbert scheme by providing new examples that are not necessarily homogeneous ideals.

Findings

New examples of elementary components are constructed.

Generalization of previous homogeneous ideal examples.

Enhanced understanding of zero-dimensional subschemes.

Abstract

Let $K$ be an algebraically closed field of characteristic 0, and let $H$ denote the Hilbert scheme of $m$ points of affine n-space $A^n$. An elementary component $E$ of $H$ is an irreducible component such that every $K$-point $[I]$ in $E$ represents a length-$m$ closed subscheme Spec$(K[x_1,\dots,x_n]/I)$ of $A^n$ that is supported at one point. Iarrobino and Emsalem gave the first explicit examples (with $m > 1$) of elementary components in ["Some zero-dimensional generic singularities: Finite algebras having small tangent space", Comp. Math. 36 (1978), pp. 145-188]; in their examples, the ideals $I$ were homogeneous (up to a change of coordinates corresponding to a translation of $A^n$). We generalize their construction to obtain new examples of elementary components.