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首页> 外文期刊>Journal of the Mathematical Society of Japan >Jacobi inversion on strata of the Jacobian of the C_(rs) curve y~r = f(x), Ⅱ
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Jacobi inversion on strata of the Jacobian of the C_(rs) curve y~r = f(x), Ⅱ

机译：C_（rs）曲线y〜r = f（x），Ⅱ的雅可比行列上的Jacobi反演

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摘要

Previous work by the authors (this journal, 60 (2008), 1009-1044) produced equations that hold on certain loci of the Jacobian of a cyclic C_(rs) curve. A curve of this type generalizes elliptic curves, and the equations in question are given in terms of (Klein's) generalization of Weierstrass' σ-function. The key tool is a matrix with entries that are polynomial in the coordinates of the affine plane model of the curve, thus can be expressed in terms of σ and its derivatives. The key geometric loci on the Jacobian of the curve give a stratification of Brill-Noether type. The results are of the type of Riemann-Kempf singularity theorem, the methods are germane to those used by J. D. Fay, who gave vanishing tables for Riemann's θ-function and its derivatives. The main objects we use were developed by several contemporary authors, aside from the classical definitions: meromorphic differentials were expressed in terms of the coordinates mainly by V. M. Buchstaber, J. C. Eilbeck, V. Z. Enolski, D. V. Leykin, and Taylor expansions for σ in terms of Schur polynomials also contributed by A. Nakayashiki, in terms of Sato's τ-function. Within this framework, following specific results for σ-derivatives given by Y. Onishi, we arrive at our main results, namely statements on the vanishing on given strata of the partial derivatives of σ indexed by Young-diagrams subsets that can be worked out in terms of the Weierstrass semigroup of the curve at its point at infinity. The combinatorial statements hold not only for Jacobians but for the stratification of Sato's infinite-dimensional Grassmann manifold as well.
机译：作者的先前工作（本期刊，60（2008），1009-1044）产生了方程，这些方程保持在循环C_（rs）曲线的雅可比行列的某些位点上。这种类型的曲线可以推广椭圆曲线，并且根据Weierstrass的σ函数的（Klein's推广）给出相关方程。关键工具是在曲线的仿射平面模型的坐标中具有多项式的项的矩阵，因此可以用σ及其导数表示。曲线的雅可比行列上的关键几何轨迹给出了Brill-Noether类型的分层。结果是Riemann-Kempf奇异性定理的类型，这些方法与J.D.Fay使用的方法密切相关，后者给出了Riemannθ函数及其导数的消失表。除经典定义外，我们使用的主要对象是由几位当代作者开发的：亚纯微分分别由坐标表示，分别由VM Buchstaber，JC Eilbeck，VZ Enolski，DV Leykin和针对Schur的σ的泰勒展开式表示。就Sato的τ函数而言，A。Nakayashiki也提供了多项式。在此框架内，遵循Y. Onishi给出的σ导数的特定结果，我们得出了主要结果，即关于由Young-diagrams子集索引的σ偏导数在给定层上消失的声明，可以在曲线的Weierstrass半群在无穷大点处的项。组合陈述不仅适用于雅各布主义者，而且适用于佐藤的无限维格拉斯曼流形的分层。

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