Riemann复曲面、代数复曲面和Jacobiennes曲面
Surfaces de Riemann compactes, courbes algébriques complexes et leurs Jacobiennes
摘要：
拓扑上，一个属$g$的紧致黎曼曲面$X$是一个有$g$洞的环面（一个有$g$把手的球面）。本文介绍紧致黎曼曲面和代数曲线的理论。它提出的基本思想和性质作为一个简要的文章，探讨了一些他们的众多后果，并给出了一个简明的基本方面的不同观点的曲线理论。我们讨论和证明最直观的一些几何拓扑方面的代数函数和相关的黎曼曲面。定义了$X$的阿贝尔微分、正规微分、Riemann双线性关系和周期矩阵，并给出了一些推论。$X$上的全纯1-形式空间的维数为$g$作为复向量空间。给出并证明了紧致黎曼曲面上因子的基本结果。Riemann-Roch定理在紧致Riemann曲面的代数几何理论中具有极其重要的意义。它告诉我们有多少线性无关的亚纯函数在极点上有某些限制。我们提出了一个简单的直接证明这个定理，并探讨其众多的后果。我们还给出了Riemann-Hurwitz公式的一个解析证明。作为应用，我们计算了一些有趣的代数曲线的亏格。阿贝尔定理通过约数在雅可比矩阵中的像对约数进行分类。雅可比反演问题是问我们是否可以找到一个除数，它是雅可比矩阵中任意一点的原像。在第一个附录中，我们直观而明确地介绍了椭圆和超椭圆黎曼曲面。在附录二中，我们根据需要研究了结式和判别式的一些结果。
英文摘要：
Topologically, a compact Riemann surface $X$ of genus $g$ is a $g$-holed torus (a sphere with $g$ handles). This paper is an introduction to the theory of compact Riemann surfaces and algebraic curves. It presents the basic ideas and properties as an expository essay, explores some of their numerous consequences and gives a concise account of the elementary aspects of different viewpoints in curve theory. We discuss and prove most intuitively some geometric-topological aspects of the algebraic functions and the associated Riemann surfaces. Abelian and normalized differentials, Riemann's bilinear relations and the period matrix for $X$ are defined and some consequences drawn. The space of holomorphic 1-forms on $X$ has dimension $g$ as a complex vector space. Fundamental results on divisors on compact Riemann surfaces are stated and proved. The Riemann-Roch theorem is of utmost importance in the algebraic geometric theory of compact Riemann surfaces. It tells us how many linearly independent meromorphic functions there are having certain restrictions on their poles. We present a simple direct proof of this theorem and explore some of its numerous consequences. We also give an analytic proof of the Riemann-Hurwitz formula. As an application, we compute the genus of some interesting algebraic curves. Abel's theorem classifies divisors by their images in the jacobian. The Jacobi inversion problem askes whether we can find a divisor that is the preimage for an arbitrary point in the jacobian. In the first appendix, we introduced intuitively and explicitly elliptic and hyperelliptic Riemann surfaces. In the second appendix, we study some results of resultant and discriminant as needed in the paper.
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