Irregular and singular loci of commuting varieties

TL;DR

This paper investigates the structure of commuting varieties in noncommutative reductive Lie algebras, establishing bounds on singular and irregular loci, and proving the variety's rationality.

Contribution

It demonstrates the containment of the singular locus within the irregular locus, computes the codimension of the irregular locus, and proves the rationality of the commuting variety.

Findings

Singular locus is contained in the irregular locus

One irreducible component of the irregular locus has codimension 4

The commuting variety is rational

Abstract

We prove that the singular locus of the commuting variety of a noncommutative reductive Lie algebra is contained in the irregular locus and we compute the codimension of the latter. We prove that one of the irreducible components of the irregular locus has codimension 4. This yields the lower bound of the codimension of the singular locus, in particular, implies that it is at least 2. We also prove that the commuting variety is rational.