On the variety of almost commuting nilpotent matrices

TL;DR

This paper investigates the structure of matrices with commutators of rank at most one, identifying their irreducible components and relating them to the Hilbert scheme of points.

Contribution

It characterizes the irreducible components of the variety of matrices with low-rank commutators and connects them to the Hilbert scheme of points.

Findings

Identified all irreducible components of the variety.

Connected components to pairs of commuting matrices and rank-one commutator pairs.

Provided a map to the zero fiber of the Hilbert scheme of points.

Abstract

We study the variety of n by n matrices with commutator of rank at most one. We describe its irreducible components; two of them correspond to the pairs of commuting matrices, and n-2 components of smaller dimension corresponding to the pairs of rank one commutator. In our proof we define a map to the zero fiber of the Hilbert scheme of points and study the image and the fibers.