Versality of BrillNoether flags and degeneracy loci of twicemarked curves
Abstract
A line bundle on a smooth curve $C$ with two marked points determines a rank function $r(a,b) = h^0(C, L(apbq))$. This paper studies BrillNoether 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 BrillNoether loci $W^r_d(C)$ as well as BrillNoether loci with imposed ramification. For general $(C,p,q)$ we determine the dimension, singular locus, and intersection class of BrillNoether 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 GiesekerPetri theorem.
 Publication:

arXiv eprints
 Pub Date:
 March 2021
 DOI:
 10.48550/arXiv.2103.10969
 arXiv:
 arXiv:2103.10969
 Bibcode:
 2021arXiv210310969P
 Keywords:

 Mathematics  Algebraic Geometry
 EPrint:
 28 pages. v2: additional references, minor corrections