  Richardson varieties play an important role in intersection theory and in the
geometric interpretation of the Littlewood-Richardson Rule for flag varieties.
We discuss three natural generalizations of Richardson varieties which we call
projection varieties, intersection varieties, and rank varieties. In many ways,
these varieties are more fundamental than Richardson varieties and are more
easily amenable to inductive geometric constructions. In this paper, we study
the singularities of each type of generalization. Like Richardson varieties,
projection varieties are normal with rational singularities. We also study in
detail the singular loci of projection varieties in Type A Grassmannians. We
use Kleiman's Transversality Theorem to determine the singular locus of any
intersection variety in terms of the singular loci of Schubert varieties. This
is a generalization of a criterion for any Richardson variety to be smooth in
terms of the nonvanishing of certain cohomology classes which has been known by
some experts in the field, but we don't believe has been published previously.
