%AAnderson, Dave%AChen, Linda%ATarasca, Nicola%BJournal Name: International Mathematics Research Notices; Journal Volume: 2022; Journal Issue: 16
%D2021%I
%JJournal Name: International Mathematics Research Notices; Journal Volume: 2022; Journal Issue: 16
%K
%MOSTI ID: 10405706
%PMedium: X
%Tđž-classes of BrillâNoether Loci and a Determinantal Formula
%XAbstract We compute the Euler characteristic of the structure sheaf of the BrillâNoether locus of linear series with special vanishing at up to two marked points. When the BrillâNoether number $\rho $ is zero, we recover the Castelnuovo formula for the number of special linear series on a general curve; when $\rho =1$, we recover the formulas of Eisenbud-Harris, Pirola, and ChanâMartĂnâPfluegerâTeixidor for the arithmetic genus of a BrillâNoether 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 BilleyâJockuschâStanley expressing Schubert polynomials for 321-avoiding permutations as generating functions for flagged skew tableaux.
%0Journal Article