Meromorphic functions on Riemann surfaces, Weierstrass points
Project/Area Number 
10440051

Research Category 
GrantinAid for Scientific Research (B)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Basic analysis

Research Institution  Yamaguchi University 
Principal Investigator 
KATO Takao Yamaguchi University, Faculty of Science, Professor, 理学部, 教授 (10016157)

CoInvestigator(Kenkyūbuntansha) 
HOMMA Masaaki Kanagawa University, Faculty of Engineering, Professor, 工学部, 教授 (80145523)
YANAGIHARA Hiroshi Yamaguchi University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (30200538)
MASUMOTO Makoto Yamaguchi University, Faculty of Science, Associate Professor, 理学部, 助教授 (50173761)
OHBUCHI Akira Tokushima University, Faculty of Arts and Science, Associate Professor, 総合科学部, 助教授 (10211111)
SHIBA Masakazu Hiroshima University, Faculty of Engineering, Professor, 工学部, 教授 (70025469)
山田 陽 東京学芸大学, 教育学部, 助教授 (60126331)
木内 功 山口大学, 理学部, 助教授 (30271076)

Project Period (FY) 
1998 – 1999

Project Status 
Completed (Fiscal Year 1999)

Budget Amount *help 
¥6,100,000 (Direct Cost: ¥6,100,000)
Fiscal Year 1999: ¥2,900,000 (Direct Cost: ¥2,900,000)
Fiscal Year 1998: ¥3,200,000 (Direct Cost: ¥3,200,000)

Keywords  compact Riemann surface / algebraic curve / meromorphic function / gonality / BrillNoether theory / normal generation / Weierstrass point / Weierstrass 点 / リーマン面 / ブリル・ネーター理論 
Research Abstract 
In this project, we studied classification problems of compact Riemann surfaces through the existence of meromorphic functions on them. It is one of the central subject on the theory compact Riemann surfaces. The main results we obtained are the following : (a) Let k be a prime number. Assume C is a compact Riemann surface of genus g which is a ksheeted covering of a compact Riemann surface of genus h>0. In this case, we studied the structure of a subvariety WィイD31(/)dィエD3(C) of the Jacobian variety J(C). Then, we have : "If g and d are large enough comparable to k and h, then WィイD31(/)dィエD3(C) is reduced and irreducible". (As a matter of fact, we have quantitative estimates for g and d). (b) For a compact Riemann surface C of genus g, H. Martens proved that dim WィイD3r(/)dィエD3(C)【less than or equal】d2r for d【less than or equal】g+r1. Then, Mumford (resp. Keem) gave a characterization of WィイD3r(/)dィエD3(C) which sarisfies dim WィイD3r(/)dィエD3(C)=d2r (resp. d2r1) for d【less than or equal
… More
】g+r3 (resp. g+r5). If C is of odd gonality, then it is known that dim WィイD3r(/)dィエD3(C)【less than or equal】d3r. In 1996, G. Martens gave a characterization of WィイD3r(/)dィエD3(C) which satisfies dim WィイD3r(/)dィエD3(C)=d3r. As an extension of this result, we gave a characterization of WィイD3r(/)dィエD3(C) which satidfies dim WィイD3r(/)dィエD3(C)=d3r1. (c) Let L be a very ample line bundle of degree d and dimension r on a compact Riemann surface of genus g. We studied the normally generatedness of L in case d is near to g. As a result, we succeeded to describe L which fails to be normally generated in case d=2g2, 2g3. Moreover, we showed that if L is special, it is always normally generated in case d=2g5. In case d=2g6, there is one exceptional case, in which L fails to be normally generated. In our discussion, a critetion for L to be normally generated due to GreenLazarsfeld plays an important role. Finally, we thought about the case that L contributes the Clifford index, while GreenLazarsfeld did not treated this case. Less

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