1999 Fiscal Year Final Research Report Summary
Study of function fields with particular properties, and its application to coding theory
| Project/Area Number |
10640048
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Kanagawa University |
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Principal Investigator |
HOMMA Masaaki Kanagawa Univ. Mathematics, Prof., 工学部, 教授 (80145523)
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| Co-Investigator(Kenkyū-buntansha) |
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)
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| Project Period (FY) |
1998 – 1999
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| Keywords | Algebraic Curve / Special Divisor / Gonality / Weierstrass pair / Error correcting Code / AG Code |
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| 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 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.
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