Study of function fields with particular properties, and its application to coding theory

本研究的最初目的是研究具有特殊性质的代数曲线，并将其成果应用于编码理论。众所周知，每一条光滑曲线都完全由其函数域决定或刻画。因此，我们可以用光滑曲线上的函数来描述光滑曲线的性质，如：一致性，Clifford指数，Luroth半群，Weierstrass间隙等。奇异曲线上的特殊因子; 2.光滑平面曲线上特殊因子的分布; 3.代数几何在编码理论中的应用。第一个是试图发展奇异曲线的特殊因子理论。作为第一步，我们处理了在射影直线上具有2次态射的奇异曲线，在第二个方向，我们给出了位于MaxNoether边界上的光滑平面曲线上的线丛的特征的一个新的证明。证明的思想是基于无基点铅笔技巧的一种变形，利用类似的思想，我们能够证明更多的事实，最后一个方向的大部分结果都是Weierstrass对理论的应用.

The initial purpose of this research was to study algebraic curves with distinctive properties and to apply its fruit to coding theory. As is well known, each smooth curve is completely determined or characterized by its function field. Therefore we can describe specialties of smooth curves in terms of functions on these curves, like gonality, Clifford index, Luroth semigroup, Weierstrass gaps and so on. That is a paraphrase of the first part of the title of the project.Now we list the topics which we have concerned with :1. Special divisors on a singular curve ;2. Distribution of special divisors on a smooth plane curves ;3. Application of algebraic geometry to coding theory.The first one is an attempt to develop a theory of special divisor for singular curves. As the first step of the trial, we have treated singular curves which have a morphism of degree 2 onto projective line.In the second direction, we give a new proof of the characterization of line bundles on a smooth plane curve lying on Max Noether's boundary. The idea of the proof is based on a variation of a base-point-free pencil trick, and we were able to show further facts by using a similar idea.The most of results in the last direction have been gotten as applications of the theory of Weierstrass pairs.