Document Type

Article

Publication Date

4-26-2021

Published In

International Mathematics Research Notices

Abstract

We compute the Euler characteristic of the structure sheaf of the Brill–Noether locus of linear series with special vanishing at up to two marked points. When the Brill–Noether number ρ is zero, we recover the Castelnuovo formula for the number of special linear series on a general curve; when ρ=1⁠, we recover the formulas of Eisenbud-Harris, Pirola, and Chan–Martín–Pflueger–Teixidor for the arithmetic genus of a Brill–Noether curve of special divisors. These computations are obtained as applications of a new determinantal formula for the K-theory class of certain degeneracy loci. Our degeneracy locus formula also specializes to new determinantal expressions for the double Grothendieck polynomials corresponding to 321-avoiding permutations and gives double versions of the flagged skew Grothendieck polynomials recently introduced by Matsumura. Our result extends the formula of Billey–Jockusch–Stanley expressing Schubert polynomials for 321-avoiding permutations as generating functions for flagged skew tableaux.

Comments

This work is a preprint that is freely available from arXiv.org at arXiv:1705.02992, courtesy of Oxford University Press. The final publication version can be found on the publisher's website.

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