Project/Area Number  10640048 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Algebra

Research Institution  Kanagawa University 
Principal Investigator 
HOMMA Masaaki Kanagawa Univ. Mathematics, Prof., 工学部, 教授 (80145523)

CoInvestigator(Kenkyūbuntansha) 
OHBUCHI Akira Tokushima Univ. Mathematics, Ass.Prof., 総合科学部, 助教授 (10211111)
KOMEDA Jiryo Kanagawa Inst.Tech. Mathematics, Prof., 工学部, 教授 (90162065)
KATO Takao Yamaguchi Univ. Mathematics, Prof., 理学部, 教授 (10016157)

Project Fiscal Year 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥1,600,000 (Direct Cost : ¥1,600,000)
Fiscal Year 1999 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1998 : ¥900,000 (Direct Cost : ¥900,000)

Keywords  Algebraic Curve / Special Divisor / Gonality / Weierstrass pair / Error correcting Code / AG Code / 代数曲線 / 特殊因子 / 誤まり訂正符号 / 代数幾何符号 / 誤まり訂正符合 / Weierstrass点 / 特異因子 / 符号 
Research Abstract 
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 basepointfree 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.
