Singularities of Generalized Richardson Varieties

Abstract

Abstract 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 article, 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. Key Words: Homogeneous varietiesRichardson varietiesSchubert varietiesSingularities2000 Mathematics Subject Classification: Primary 14M1514N3532M10 ACKNOWLEDGMENTS We would like to thank Dave Anderson, Lawrence Ein, Sándor Kovács, Max Lieblich, Stephen Mitchell, and Lauren Williams for helpful and stimulating discussions. We thank the anonymous referees for their help in improving the exposition. We are grateful to the American Mathematics Institute, where this project was started, for providing a stimulating work environment, and to the organizers of the Localization Techniques in Equivariant Cohomology Workshop, namely, William Fulton, Rebecca Goldin, and Julianna Tymoczko. During the preparation of this article, the first author was supported by the National Science Foundation (NSF) grant DMS-0800978, and the second author was partially supported by the NSF grant DMS-0737581, NSF CAREER grant DMS-0950951535, and an Alfred P. Sloan Foundation Fellowship. Notes 1Shortly after this article was posted, Knutson, Lam, and Speyer posted a preprint [Citation31] independently proving Theorem 1.1 using Frobenius splitting techniques and a fairly intricate degeneration argument. Our proof only uses standard statements about vanishing of higher cohomology, hence is more widely applicable. Communicated by L. Ein.