Exponentiation of commuting nilpotent varieties

Abstract: Let $H$ be a linear algebraic group over an algebraically closed field of characteristic $p>0$. We prove that any "exponential map" for $H$ induces a bijection between the variety of $r$-tuples of commuting $[p]$-nilpotent elements in $Lie(H)$ and the variety of height $r$ infinitesimal one-parameter subgroups of $H$. In particular, we show that for a connected reductive group $G$ in pretty good characteristic, there is a canonical exponential map for $G$ and hence a canonical bijection between the aforementioned varieties, answering in this case questions raised both implicitly and explicitly by Suslin, Friedlander, and Bendel.