Relative Richardson Varieties

A Richardson variety is an intersection of two Schubert varieties defined by transverse flags in a vector space. Richardson varieties have many nice geometric properties; for example, a theorem of Knutson, Woo, and Yong shows that their singularities are completely determined by those of Schubert varieties. I will discuss a generalization of this theorem to a relative context, where the two transverse flags are replaced by a moving pair of flags in a vector bundle that become non-transverse at some points. I will also discuss a theorem describing the cohomology of the resulting relative Richardson variety. I will describe an application to Brill-Noether theory, and some related conjectures. This is joint work with Melody Chan.