Relative Bott-Samelson varieties

TL;DR

This paper generalizes Brion's resolution of Richardson varieties to the relative setting, showing that products of relative Bott-Samelson varieties resolve singularities of relative Richardson varieties, with applications to Brill-Noether varieties on elliptic curves.

Contribution

It introduces a relative version of Bott-Samelson varieties that resolve singularities of relative Richardson varieties, extending known results to a broader geometric context.

Findings

Resolution of singularities for relative Richardson varieties.

Extension of Brion's resolution to the relative setting.

Application to Brill-Noether varieties with ramification on elliptic curves.

Abstract

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.