2007 Fiscal Year Final Research Report Summary
Various aspects of pencils of algebraic curves
| Project/Area Number |
16340008
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | Osaka University |
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Principal Investigator |
KONNO Kazuhiro Osaka University, Graduate School of Scence, Professor (10186869)
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| Co-Investigator(Kenkyū-buntansha) |
USUI Sampei Osaka University, Graduate School of Science, Professor (90117002)
ENDO Hisaaki Osaka University, Graduate School of Science, Associate Professor (20323777)
ASHIKAGA Tadashi Tohtoku Gakain University, Facultyof Engineering, Professor (90125203)
TOKUNAGA Hiro-o Tokyo Metrophtan University, Graduate School of Science, Professor (30211395)
SHIMADA Ichiro Hokkaido University, Graduate School of Saence, Associate Professor (10235616)
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| Project Period (FY) |
2004 – 2007
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| Keywords | Pencil of algebraic curves / local signature / branched covering / Mordell-Weil lattice / Galois point / Zanski pair |
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| Research Abstract |
The purpose of the research project is to study pencils of algebraic curves from mainly two sides. One is algebraic geometry and the other is topology. The interaction between two fields of mathematics has been crucial .We paid our special attention to the following topics.
(1) The slope of fibred algebraic surfaces,
(2) The localization of signature,
(3) The Mordell-Weil lattices.
Our achievements include: the lower bounds of the slopes for fibred rational surfaces, the inequality involving the Clifford index of a general fibre and the Mordell Weil rank, the complete description of fibred rational surfaces with extremal Mordell-Weil rank, the rationality or ellipticity of the singularity obtained by contracting the fixed part of the relative canonial linear system for a normal surface singularity and a fibred algebraic surface, the comparison between local signatures obtained from the eta invariant and semi-stable reduction of a singular fibre germ. During the research period, in total, 57 papers are written and 121 research presentations are given by our investigators. We organized (or co-organized) 6 conferences. Above all, we held the international conference "Algebraic Geometry in East Asia, II" at Hanoi(Vietnam), October 2005, in order to promote research interaction among algebraic geometers in the area. We published the proceedings of the meetings to show the activity of the research project. Also, we invited 6 researchers in abroad and discussed verious problems fruitfully.
In sum, our purpose has been achieved m great success, thanks to the active contributions and cooperation of our joint investigators.
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