Relative Bott–Samelson Varieties

We prove that, defined with respect to versal flags, the product of two relative Bott–Samelson varieties over a 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. As an application, this gives a resolution of singularities, with a modular interpretation, for the Brill–Noether variety with imposed ramification on twice-marked elliptic curves.