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

A line bundle on a smooth curve $C$ with two marked points determines a rank function $r(a,b) = h^0(C, L(-ap-bq))$. This paper studies Brill-Noether degeneracy loci; such a locus is defined to be the closure in $\operatorname{Pic}^d(C)$ of the locus of line bundles with a specified rank function. 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 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 $\operatorname{Pic}^d(C)$, conjectured by Melody Chan and the author. We prove this versality theorem by showing the injectivity of a generalized Petri map, along the lines of Eisenbud and Harris's proof of the Gieseker-Petri theorem.