Versality of Brill–Noether flags and degeneracy loci of twice-marked curves

A Brill-Noether degeneracy locus is the closure in Pic d (C) of the locus of line bundles with a specified rank function r(a, b) = h 0 (C, L(-ap -bq)).These loci generalize the classical Brill-Noether loci W r d (C) as well as Brill-Noether loci with imposed ramification.For general (C, p, q), we determine the dimension, singular locus, and intersection class of Brill-Noether degeneracy loci, generalizing classical results about W r d (C).The intersection class has a combinatorial interpretation in terms of the number of reduced words for a permutation associated with the rank function, or alternatively the number of saturated chains in the Bruhat order.The essential tool is a versality theorem for a certain pair of flags on Pic d (C), conjectured by Melody Chan and the author.