  We prove that, defined with respect to versal flags, the product of two
relative Bott-Samelson varieties over the flag bundle is a resolution of
singularities of a relative Richardson variety. This result generalizes Brion's
resolution of singularities of Richardson varieties to the relative setting. It
reflects the phenomenon that the local geometry of a relative Richardson
variety is completely governed by the two intersecting relative Schubert
varieties, studied by Chan-Pflueger. We also prove an analogous theorem in the
case of relative Grassmannian Richardson varieties, thereby furnishing a
resolution of singularities for the Brill-Noether variety with imposed
ramification on twice-marked elliptic curves.
