Meromorphic functions on compact Riemann surfaces

利用紧黎曼曲面上亚纯函数和共形不变量的存在性，研究了紧黎曼曲面的分类问题。设C为g属的紧致黎曼曲面，W^γ_d(C)为雅可比变换J (C)中由阶数为d、维数为γ的有效因子像组成的子变换。1992年Coppens-Kim-Martens证明了如果C的gonality on(C)是奇的，那么dim W^γ_d(C)小于等于d - 3γ对任意d小于等于g - 1都成立。1996年，Martens给出了当W^γ_d(C) = d - 3γ满足d <≤g - 2时，C和W^γ_d(C)的一个表征。1999年，Kato-Keem给出了当W^γ_d(C) = d - 3γ - 1，且d <≤g - 4时，C和W^γ_d(C)的一个表征。这是我国科学研究资助项目(B)(2),（2000-2001）“关于亚纯函数和Weierstrass点的研究”#10440051的主要研究成果之一。在本研究中，我们首先注意到，即使在……的情况下，当C不对合时，暗淡的W^γ_d(C) <或等于d - 3γ也成立。然后，给出了弱W^γ_d(C) = d - 3γ， d【小于或等于】g - 2和弱W^γ_d(C) = d - 3γ - 1， d【小于或等于】g - 4.2时C和W^γ_d(C)的表征。我们研究与纠错编码理论有关的射影系统，特别是代数几何编码。设S是有限域上有q个元素的射影空间中的集合。然后，我们证明了S是2的并集。3)与超平面相交的点的个数满足一定条件的子空间。这是对“本马-金-柳”结果的改进。研究了光滑平面曲线上总拐点对费马曲线的刻画。d次费马曲线共有3d个拐点。我们考虑了相反的情况，并证明了如果一个光滑的5次平面曲线具有5阶和15阶的自同构拐点，那么它与费马曲线是等价的。少

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