Components of Hilbert Schemes of low degree and smoothable algebras

TL;DR

This paper characterizes the irreducible components of Hilbert schemes for 9 and 10 points in affine space, proving smoothability of certain local algebras and relating cactus and secant Grassmann varieties.

Contribution

It provides a detailed description of Hilbert scheme components for low degrees and proves smoothability for specific local algebras, connecting algebraic and geometric properties.

Findings

Irreducible components of Hilbert schemes for d=9,10 are described.

Finite local algebras of degrees 9,10 with socle dimension 2 are smoothable.

Equality of cactus Grassmann and secant Grassmann varieties established.

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.