2004 Fiscal Year Final Research Report Summary
Special linear systems on compact Riemann surfaces
| Project/Area Number |
15540173
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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) |
MASUMOTO Makoto Yamaguchi University, Faculty of Science, Professor, 理学部, 教授 (50173761)
YANAGI Kenjiro Yamaguchi University, Faculty of Engineering, Professor, 工学部, 教授 (90108267)
YANAGIHARA Hiroshi Yamaguchi University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (30200538)
HOMMA Masaaki Kanagawa University, Faculty of Engineering, Professor, 工学部, 教授 (80145523)
OBUCHI Akira Tokushima University, Faculty of Integrated Arts and Sciences, Professor, 総合科学部, 教授 (10211111)
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| Project Period (FY) |
2003 – 2004
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| Keywords | compact Riemann surfaces / algebraic curves / meromorphic functions / linear systems / 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^r_d(C) be a subvariety which consists of the image of effective divisors of degree d and dimension r 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^r_d(C) 【less than or equal】 d - 3r holds for any d 【less than or equal】 g - 1. In 1996, Martens gave a characterization of C and W^r_d(C) in case dim W^r_d(C) = d - 3r with d 【less than or equal】 g - 2. In 1999, Kato-Keem gave a characterization of C and W^r_d(C) in case dim W^r_d(C) = d - 3r - 1 with d 【less than or equal】 g - 4. In 2001, Kato remarked that even in the case gon(C) is even, if C doesn't have an involution, dim W^r_d(C) 【less than or equal】 d - 3r holds, too. Then, one has a characterization of C and W^r_d(C) in cases dim W^r_d(C) = d - 3r with d 【less than or equal】 g - 2 and dim W^r_d(C) = d - 3r - 1 with 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 (C)(2), (2000-2001) entitled "A study on meromorphic functions on compact Riemann surfaces" #12640180. In this project, we almost succeeded a characterization of C and W^r_d(C) in cases dim W^r_d(C) = d - 3r - 2.
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) is equal to the Griesmer bound minus 1.
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Research Products
(11 results)
