Commuting matrices and the Hilbert scheme of points on affine spaces

TL;DR

This paper provides algebraic and monadic descriptions of the Hilbert scheme of points on affine spaces, extending Nakajima's work, and proves irreducibility of the scheme for up to 10 points in three dimensions.

Contribution

It introduces new descriptions of the Hilbert scheme on affine spaces and applies recent results on commuting matrices to establish irreducibility for certain cases.

Findings

Provides algebraic and monadic descriptions of the Hilbert scheme

Extends Nakajima's representation to higher dimensions

Shows irreducibility of the scheme for c ≤ 10 in three dimensions

Abstract

We give a linear algebraic and a monadic descriptions of the Hilbert scheme of points on the affine space of dimension $n$ which naturally extends Nakajima's representation of the Hilbert scheme of points on the plane. As an application of our ideas and recent results from the literature on commuting matrices, we show that the Hilbert scheme of $c$ points on $(\C^3)$ is irreducible for $c\le 10$.