2001 Fiscal Year Final Research Report Summary
Meromorphic functions on compact Riemann surfaces
| Project/Area Number |
12640180
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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 |
Basic analysis
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| Research Institution | Yamaguchi University |
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Principal Investigator |
KATO Takao Yamaguchi University, Faculty of Science, Professor, 理学部, 教授 (10016157)
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| Co-Investigator(Kenkyū-buntansha) |
YANAGIHARA Hiroshi Yamaguchi University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (30200538)
YANAGI Kenjiro Yamaguchi University, Faculty of Engineering, Professor, 工学部, 教授 (90108267)
MASUMOTO Makoto Yamaguchi University, Faculty of Science, Associate Professor, 理学部, 助教授 (50173761)
OBUCHI Akira Tokushima University, Faculty of Integrated Arts and Sciences, Associate Professor, 総合科学部, 助教授 (10211111)
HOMMA Masaaki Kanagawa University, Faculty of Engineering, Professor, 工学部, 教授 (80145523)
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| Project Period (FY) |
2000 – 2001
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| Keywords | compact Riemann surfaces / algebraic curves / meromorphic functions / gonality / error-correcting coding theory |
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| Research Abstract |
We study classification problems for compact Riemann surfaces through the existence of meromorphic functions on them and conformal invariants.
1. Let C be a compact Riemann surface of genus g and W^γ_d(C) be a subvariety which consists of the image of effective divisors of degree d and dimension γ in the Jacobian variety J (C). In 1992 Coppens-Kim-Martens proved that if the gonality gon(C) of C is odd, then dim W^γ_d(C) 【less than or equal】 d - 3γ holds for any d 【less than or equal】 g - 1. In 1996, Martens gave a characterization of C and W^γ_d(C) in case dim W^γ_d(C) = d - 3γ held for d 【less than or equal】 g - 2. In 1999, Kato-Keem gave a characterization of C and W^γ_d(C) in case dim W^γ_d(C) = d - 3γ - 1 held for d 【less than or equal】 g - 4. It is one of the main results in our study supported by Grant-in -Aid for Scientific Research (B)(2), (2000-2001) entitled "A study on meromorphic functions and Weierstrass points" #10440051. In the present study, first, we remark that even in
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the case gon(C) is even, if C doesn't have an involution, dim W^γ_d(C) 【less than or equal】 d - 3γ holds, too. Then, we give a chracterization of C and W^γ_d(C) in case dim W^γ_d(C) = d - 3γ, d 【less than or equal】 g - 2 and dim W^γ_d(C) = d - 3γ - 1, d 【less than or equal】 g - 4.
2. We study projective systems concerned with the error-correcting coding theory, in particular the algebraic geometry codes. Let S be a set in the projective space over a finite field with q elements. Then, we prove that S is a union of 2 (resp. 3) subspaces provided that the numbers of points which intersect with hyperplanes satisfy some conditions. It is an improvement of a result of Homma-Kim-Yoo.
We study a characterization of the Fermat curve in terms of total inflection points on smooth plane curve. The Fermat curve of degree d has 3d total inflection points. We consider the converse and prove that if a smooth plane curve of degree 5 has an automorphism of order 5 and 15 total inflection points, then it is birationally equivalent to the Fermat curve. Less
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