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 $\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 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.

中文翻译：

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𝐾-类 Brill-Noether 位点和一个行列式

我们计算了线性级数的 Brill-Noether 轨迹的结构层的欧拉特征，在最多两个标记点处具有特殊消失。当 Brill-Noether 数 $\rho $ 为零时，我们恢复了一般曲线上特殊线性级数数的 Castelnuovo 公式；当 $\rho =1$ 时，我们恢复了 Eisenbud-Harris、Pirola 和 Chan-Martín-Pflueger-Teixidor 的公式，用于特殊因数的 Brill-Noether 曲线的算术属。这些计算是作为某些退化位点的$K$-theory 类的新行列式公式的应用而获得的。我们的简并轨迹公式还专门用于对应于 321 避免排列的双格洛腾迪克多项式的新行列式表达式，并给出了松村最近引入的带标志的倾斜格洛腾迪克多项式的双版本。我们的结果扩展了 Billey-Jockusch-Stanley 的公式，将 321 避免排列的舒伯特多项式表示为标记倾斜画面的生成函数。