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

A Brill-Noether degeneracy locus is closure in ${\text{Pic}}^d(C)$ 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 $W_d^r(C)$ 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 $W_d^r(C)$. The intersection class has a combinatorial interpretation in terms of the number of reduced words for a permutation associated to 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 ${\text{Pic}}^d(C)$, conjectured by Melody Chan and the author.