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

TL;DR

This paper generalizes classical Brill-Noether loci by studying degeneracy loci of line bundles with specified rank functions on curves, providing dimension, singularity, and intersection class results.

Contribution

It introduces a versality theorem for pairs of flags on Picard varieties, enabling the analysis of Brill-Noether degeneracy loci and their combinatorial interpretations.

Findings

Determined the dimension and singular locus of Brill-Noether degeneracy loci.

Computed the intersection class with a combinatorial interpretation.

Established a versality theorem for flags on Picard varieties.

Abstract

A Brill-Noether degeneracy locus is 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 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 $\Pic^d(C)$, conjectured by Melody Chan and the author.