Some elementary components of the Hilbert scheme of points

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.