Modules with small Loewy length

A module of complexity c for E≅(Z/p)r in characteristic p has Loewy length at least (p−1)(r−c)+1. We study the case of equality. If p is odd, the only rank varieties possible are finite unions of linear subspaces of dimension c , and every such rank variety occurs. If p=2, the variety has to be equidimensional. If such a variety is a finite union of set theoretic complete intersections then it occurs for such a module, but otherwise the situation is unclear. Exterior algebras in any characteristic are also treated, and follow the same behaviour as the case p=2 above.