Special linear systems on Riemann surfaces

通过亚纯函数和共形不动点的存在性研究了紧致黎曼曲面的分类问题.设C是亏格g的紧致Riemann曲面。上的铅笔的最小次数称为C的角性，记为gon（C）。这个量是共形不变量。众所周知，2 [小于或等于] gon（C）[小于或等于] [（g + 3）/2]。另一方面，设s_2（C）是C上简单网的最小度。当s_2（C）满足（3 +<$<8 g +1>）/2 [小于或等于]s_2（C）[小于或等于] g + 2时，这两个量密切相关。事实上，如果gon（C）= 2，则s_2（C）= g + 2，反之亦然。当g ≥ 6时，C是椭圆-超椭圆的当且仅当s_2（C）= g + 1.在这个项目中，我们证明了对几乎所有的g，不存在C使得s_2（C）= g。此外，当C是亏格9的4-阶数，且具有涡卷不变量（4，1，1）时，我们确定s_2（C）.这个病例似乎是四个病例中最复杂的一个。设F_q是具有q个元素的有限域，C F^n_q是线性[n，k，d]q码。设n_q（k，d）为固定k，d的码长中的最小值。n_q（k，d）存在一个上界，称为Griesmer界。在本项目中，我们证明了对于d的一定范围，n_q（k，d）（k = 5，6）等于Griesmer界减1。作为Weierstrass点概念的推广，可以通过选择适当的n个点来定义Weierstrass n元组的概念。利用Weierstrass n元组的纯间隔，得到了Goppa码的最小距离的一个估计。

We study classification problems for compact Riemann surfaces through the existence of meromorphic functions on them and conformal invarinats.1. Let C be a compact Riemann surface of genus g. The minimal degree of pencils on is said to be the gonality of C and denoted by gon(C). This quantity is a conformal invarinat. It is well-known that 2 【less than or equal】 gon(C) 【less than or equal】 [(g + 3)/2]. On the other hand, let s_2(C) be the minimal degree of simple nets on C. While s_2(C) satisfies (3 + √<8g+1>)/2 【less than or equal】s_2(C) 【less than or equal】 g + 2, these two quantities relate strongly. As a matter of fact, If gon(C) = 2, then s_2(C) = g + 2 and vice versa. Furthermore, in case g 【greater than or equal】 6, that C is elliptic-hyperelliptic if and only if s_2(C) = g + 1. In this project, we show that for almost all g, there is no C such that s_2(C) = g. Moreover, in the case where C is 4-gonal of genus 9 with the scrollar invariant (4,1,1), we decided s_2(C). This case seems to be the most complicated case among 4-gonal cases.2. Let F_q be a finite fields with q elements and C ⊂F^n_q be a linear [n,k,d]q code. Let n_q(k,d) be the minimum of the code lengths for fixed k, d. There is an upper bound of n_q(k,d) known as the Griesmer bound. In this project, we show that for some range of d's, n_q(k,d) (for k = 5,6) is equal to the Griesmer bound minus 1.As a generalization of the notion of the Weierstrass point, one can define the notion of Weierstrass n-tuple by choosing appropriate n points. Using the pure gaps of Weierstrass n-tuple, we obtained an estimate of the minimal distance of the Goppa codes.