Singularities of Generalized Richardson Varieties

Sara Billey,Izzet Coskun

doi:10.1080/00927872.2011.551903

Sara Billey, Izzet Coskun

Open Access

https://doi.org/10.1080/00927872.2011.551903

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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.

Highlights

Richardson varieties play an important role in intersection theory and in the geometric interpretation of the Littlewood-Richardson Rule for flag varieties
We show that a Richardson variety in the Grassmannian variety G(k, n) is smooth if and only if it is a Segre product of Grassmannians
A projection variety is the image of a Richardson variety R(u, v) under a projection πQ with its reduced induced structure

Summary

The singularities of intersection varieties and Richardson varieties

We review the necessary notation and background for this article. In particular, we recall Kleiman’s Transversality Theorem and review its application to the singular loci of Richardson varieties and intersection varieties. We conclude that the singular locus of an intersection variety is empty if and only if the cohomology classes [Xusiing] · j=i[Xuj ] = 0 for all 1 ≤ i ≤ r This concludes the proof of Proposition 2.8. Let Xusing and Xsving denote the singular loci of the two Schubert varieties Xu and Xv, respectively. Even for Richardson varieties R(u, v) in the Grassmannian, there may not be any torus fixed points in the smooth locus of R(u, v). In Remark 4.15, we will characterize the Richardson varieties in the Grassmannian that contain a torus fixed smooth point. When G/P is the Grassmannian G(k, n), Corollary 2.9 implies a nice, geometric characterization of the smooth Richardson varieties.

Projection Varieties

Singularities of Grassmannian projection varieties