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

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 $n^2$.

7 pages