# The variety of nilpotent pairs $(A,B)$ with $[A,B] = \lambda I$

## TL;DR

This paper investigates the structure of the variety of nilpotent matrix pairs with a fixed scalar commutator over an algebraically closed field of positive characteristic, establishing irreducibility and dimension results.

## Contribution

It proves that the variety of nilpotent pairs with a scalar commutator is irreducible and determines its dimension when the matrix size is a multiple of the characteristic.

## Key findings

- The variety is irreducible for matrices of size divisible by the characteristic.
- The dimension of the variety is equal to the square of the matrix size.
- Results hold over algebraically closed fields of positive characteristic.

## Abstract

Let $k$ be an algebraically closed field of characteristic $p >0$. We consider the variety of nilpotent pairs $(A,B)$ with $[A,B]=\lambda I$, namely the set of pairs $ X = \{ (A,B) \in M_n(k) \times M_n(k) \mid A,B \text{ nilpotent}, [A,B]=\lambda I, \lambda \in k \}$. We prove that if $n=pr$, then $X$ is irreducible of dimension $n^2$.