Motivated by questions in modular representation theory, Carlson, Friedlander, and the first author introduced the varieties E(r, g) of rdimensional abelian p-nilpotent subalgebras of a p-restricted Lie algebra g in [CFP2]. In this paper, we identify the varieties E(r, g) for a reductive restricted Lie algebra g and r the maximal dimension of an abelian p-nilpotent subalgebra of g.

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Let G be an infinitesimal group scheme of finite height r and V (G) the scheme which represents 1-parameter subgroups of G. We consider sheaves over the projective support variety P(G) constructed from a G-module M . We show that if P(G) is regular then the sheaf H [1] (M ) is zero if and only if M is projective. In general, H [1] defines a functor from the stable module category and we prove that its kernel is a thick triangulated subcategory. Finally, we give examples of G such that P(G) is regular and indicate, in characteristic 2, the connection to the BGG correspondence. Along the way we will provide new proofs of some known results and correct some errors in the literature.