Special linear systems on Riemann surfaces

利用紧黎曼曲面上亚纯函数的存在性和共形不变性，研究了紧黎曼曲面的分类问题。设C是亏格g的紧致Riemann曲面，称其上铅笔的最小次数为C的性度，记为gon(C)。这个量是一个共形不变式。众所周知，2[小于或等于]gon(C)[小于或等于][(G3)/2]。另一方面，设S_2(C)是C上单网的极小度，当S_2(C)满足(3√&lt；8g1&gt；)/2[小于或等于]S_2(C)[小于或等于]g2时，这两个量有很强的相关性。事实上，如果Gon(C)=2，则S_2(C)=g2，反之亦然。此外，在g[大于或等于]6时，C是椭圆-超椭圆的当且仅当S_2(C)=g。在这个项目中，我们证明了对于几乎所有的g，不存在C使得S_2(C)=g.此外，当C是具有卷轴不变量(4，1，1)的亏格为9的4-正方形时，我们决定S_2(C).这个案件似乎是四角形案件中最复杂的案件。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)是固定k，d的最小码长，n_q(k，d)有一个上界，称为Griesmer界。在这个项目中，我们证明了对于d‘s的某个范围，n_q(k，d)(当k=5，6)等于Griesmer界减1。作为weerstrass点概念的推广，我们可以通过选择适当的n点来定义weerstrass n元组的概念。利用魏尔斯特拉斯n元组的纯间隙，我们得到了Goppa码的最小距离的估计。

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.