Motivated by questions in modular representation theory,
Carlson, Friedlander and the speaker instigated the study of projective
varieties of abelian p-nilpotent subalgebras of a fixed dimension r for a
p-Lie algebra g. These varieties are closely related to the much studied
class of varieties of r-tuples of commuting p-nilpotent matrices which
remain highly mysterious when r>2.

In this talk, I shall present some of the representation-theoretic
motivation behind the study of these varieties and describe their geometry
in a very special case when it is well understood: namely, when r is the
maximal dimension of an abelian p-nilpotent subalgebra of a semisimple Lie
algebra g.

This is joint work with J. Stark.