  A Richardson variety in a flag variety is an intersection of two Schubert
varieties defined by transverse flags. We define and study relative Richardson
varieties, which are defined over a base scheme with a vector bundle and two
flags. To do so, we generalize transversality of flags to a relative notion,
versality, that allows the flags to be non-transverse over some fibers.
Relative Richardson varieties share many of the geometric properties of
Richardson varieties. We generalize several geometric and cohomological facts
about Richardson varieties to relative Richardson varieties. We also prove that
the local geometry of a relative Richardson variety is governed, in a precise
sense, by the two intersecting Schubert varieties, giving a generalization, in
the flag variety case, of a theorem of Knutson-Woo-Yong; we also generalize
this result to intersections of arbitrarily many relative Schubert varieties.
We give an application to Brill-Noether varieties on elliptic curves, and a
conjectural generalization to higher genus curves.
