Let $G$ be a reductive group with Lie algebra $\mathfrak{g}$ over a field of positive characteristic $p$. We will define the projective variety $\mathbb E(r, \mathfrak g)$ of $r$-dimensional elementary abelian subalgebras of $\mathfrak g$. The case $r = 1$ has been extensively studied; $\mathbb E(1, \mathfrak g)$ is simply the projectivization of the restricted nullcone, which we know to be the cohomology variety of $\mathfrak g$. When $r = 2$ the geometry of $\mathbb E(r, \mathfrak g)$ is known due to work of Premet. We show, with some assumptions on $p$, how to compute $\mathbb E(r, \mathfrak g)$ when $r$ is the maximal possible dimension of such a subalgebra. Very little is known about $\mathbb E(r, \mathfrak g)$ when $r$ is not extremal.

This is joint work with Julia Pevtsova.