Algebraic sets defined by the commutator matrix

TL;DR

This paper investigates algebraic sets of matrix pairs defined by conditions on their commutator matrix, establishing their algebraic properties and singularity types over fields of positive characteristic.

Contribution

It identifies systems of parameters, proves these sets are complete intersections, and classifies their singularity properties as $F$-pure and $F$-regular.

Findings

The algebraic sets are complete intersections.

They are $F$-pure over fields of positive characteristic.

The set with zero diagonal commutator is $F$-regular.

Abstract

In this paper we study algebraic sets of pairs of matrices defined by the vanishing of either the diagonal of their commutator matrix or its anti-diagonal. We find a system of parameters for the coordinate rings of these two sets and their intersection and show that they are complete intersections. Moreover, we prove that these algebraic sets are $F$-pure over a field of positive prime characteristic and the algebraic set of pairs of matrices with the zero diagonal commutator is $F$-regular.