Components of Hilbert Schemes of low degree and smoothable algebras

Abstract

In this article, we describe the irreducible components of the Hilbert scheme of $d$ points on $\mathbb{A}^n$ for $d=9,10$. The main techniques we use are the variety of commuting matrices and analyzing loci of local algebras with a specific Hilbert function. We further prove that any finite local algebra of degrees $9,10$ and the socle dimension $2$ is smoothable. As the main consequence, we establish the equality of the cactus Grassmann and the secant Grassmann variety in the corresponding cases.