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Meromorphic functions on compact Riemann surfaces

紧黎曼曲面上的亚纯函数

基本信息

批准号：
12640180
负责人：
KATO Takao
金额：
$ 2.18万
依托单位：
Yamaguchi University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

项目摘要

We study classification problems for compact Riemann surfaces through the existence of meromorphic functions on them and conformal invariants.1. Let C be a compact Riemann surface of genus g and W^γ_d(C) be a subvariety which consists of the image of effective divisors of degree d and dimension γ in the Jacobian variety J (C). In 1992 Coppens-Kim-Martens proved that if the gonality gon(C) of C is odd, then dim W^γ_d(C) 【less than or equal】 d - 3γ holds for any d 【less than or equal】 g - 1. In 1996, Martens gave a characterization of C and W^γ_d(C) in case dim W^γ_d(C) = d - 3γ held for d 【less than or equal】 g - 2. In 1999, Kato-Keem gave a characterization of C and W^γ_d(C) in case dim W^γ_d(C) = d - 3γ - 1 held for d 【less than or equal】 g - 4. It is one of the main results in our study supported by Grant-in -Aid for Scientific Research (B)(2), (2000-2001) entitled "A study on meromorphic functions and Weierstrass points" #10440051. In the present study, first, we remark that even in … More the case gon(C) is even, if C doesn't have an involution, dim W^γ_d(C) 【less than or equal】 d - 3γ holds, too. Then, we give a chracterization of C and W^γ_d(C) in case dim W^γ_d(C) = d - 3γ, d 【less than or equal】 g - 2 and dim W^γ_d(C) = d - 3γ - 1, d 【less than or equal】 g - 4.2. We study projective systems concerned with the error-correcting coding theory, in particular the algebraic geometry codes. Let S be a set in the projective space over a finite field with q elements. Then, we prove that S is a union of 2 (resp. 3) subspaces provided that the numbers of points which intersect with hyperplanes satisfy some conditions. It is an improvement of a result of Homma-Kim-Yoo.We study a characterization of the Fermat curve in terms of total inflection points on smooth plane curve. The Fermat curve of degree d has 3d total inflection points. We consider the converse and prove that if a smooth plane curve of degree 5 has an automorphism of order 5 and 15 total inflection points, then it is birationally equivalent to the Fermat curve. Less

通过亚纯函数和共形不变量的存在性研究了紧致黎曼曲面的分类问题.设C是亏格为g的紧致Riemann曲面，W^γ_d（C）是Jacobian簇J（C）中d次和维数为γ的有效因子的像所构成的子簇. 1992年，Coppens-Kim-Martens证明了如果C的角形（C）是奇数，则dim W^γ_d（C）[小于或等于] d - 3γ对任何d [小于或等于] g - 1成立。1996年，Martens给出了当dim W^γ_d（C）= d - 3γ时C和W^γ_d（C）的一个刻划，其中d [小于或等于] g - 2. 1999年，Kato-Keem给出了当dim W^γ_d（C）= d - 3γ - 1且d [小于或等于] g - 4时C和W^γ_d（C）的一个刻划.这是我们的研究的主要结果之一，资助的科学研究补助金（B）（2），（2000-2001）题为“亚纯函数和Weierstrass点的研究”#10440051。在本研究中，首先，我们注意到，即使在 ...更多信息在gon（C）为偶数的情况下，如果C没有对合，dim W^γ d（C）[小于或等于] d - 3γ也成立.在dim W^γ_d（C）= d - 3γ，d ≤ g - 2和dim W^γ_d（C）= d-3γ - 1，d ≤ g - 4.2的情形下，给出了C和W^γ_d（C）的一个特征.我们研究了与纠错编码理论有关的射影系统，特别是代数几何码。设S是q元有限域上射影空间中的集合。然后，我们证明了S是2的并（分别为）。3)子空间，条件是与超平面相交的点的数目满足某些条件。本文改进了Homma-Kim-Yoo的一个结果，利用光滑平面曲线上的全拐点来刻画Fermat曲线。d次费马曲线有3d个总拐点。我们考虑相反的情况，并证明了如果一个光滑的5次平面曲线有一个5阶自同构和15个总拐点，那么它是双有理等价的Fermat曲线。少