Abstract
Abstract 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.
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