摘要翻译：
本文将亏格g正则曲线C上的截面K作为证明奇异轨迹θ因子的几何新结果的关键工具。K因子是由一组含有C的二次曲面的线性相关的条件所刻画的，并且C上的有效因子是以θ定义的雅可比环面上特殊变体的交点数。在Andreotti-Mayer方法的框架下，给出了与Mumford同构密切相关的代数曲线模空间上的线丛截面，其零轨迹刻画了Schottky问题的特殊变体，这一结果也再现了以前唯一已知的G=4的情形。这种新方法基于全纯阿贝尔微分二重积行列式关系的组合，揭示了阿贝尔全纯微分的基本结构，并用θ函数导出了阿贝尔全纯微分的规范基和定义Mumford形式的常数的显式表达式。此外，由Siegel度量导出的正则曲线模空间上的度量仅用Riemann周期矩阵表示，该度量等价于Bergman再生核平方的Kodaira-Spencer映射，这是在一般情况下G=2和G=3已知的结果。最后，导出的Siegel体积形式用Mumford形式表示。
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英文标题：
《The Singular Locus of the Theta Divisor and Quadrics through a Canonical
  Curve》
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作者：
Marco Matone, Roberto Volpato
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最新提交年份：
2007
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分类信息：

一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Physics        物理学
二级分类：High Energy Physics - Theory        高能物理-理论
分类描述：Formal aspects of quantum field theory. String theory, supersymmetry and supergravity.
量子场论的形式方面。弦理论，超对称性和超引力。
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一级分类：Mathematics        数学
二级分类：Combinatorics        组合学
分类描述：Discrete mathematics, graph theory, enumeration, combinatorial optimization, Ramsey theory, combinatorial game theory
离散数学，图论，计数，组合优化，拉姆齐理论，组合对策论
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英文摘要：
  A section K on a genus g canonical curve C is identified as the key tool to prove new results on the geometry of the singular locus Theta_s of the theta divisor. The K divisor is characterized by the condition of linear dependence of a set of quadrics containing C and naturally associated to a degree g effective divisor on C. K counts the number of intersections of special varieties on the Jacobian torus defined in terms of Theta_s. It also identifies sections of line bundles on the moduli space of algebraic curves, closely related to the Mumford isomorphism, whose zero loci characterize special varieties in the framework of the Andreotti-Mayer approach to the Schottky problem, a result which also reproduces the only previously known case g=4. This new approach, based on the combinatorics of determinantal relations for two-fold products of holomorphic abelian differentials, sheds light on basic structures, and leads to the explicit expressions, in terms of theta functions, of the canonical basis of the abelian holomorphic differentials and of the constant defining the Mumford form. Furthermore, the metric on the moduli space of canonical curves, induced by the Siegel metric, which is shown to be equivalent to the Kodaira-Spencer map of the square of the Bergman reproducing kernel, is explicitly expressed in terms of the Riemann period matrix only, a result previously known for the trivial cases g=2 and g=3. Finally, the induced Siegel volume form is expressed in terms of the Mumford form.
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PDF链接：
https://arxiv.org/pdf/0710.2124