2023 Fiscal Year Final Research Report
Study of the geography of fibrations through branched coverings, differencial equations and moduli spaces
| Project/Area Number |
18K03264
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 11010:Algebra-related
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| Research Institution | Tohoku Gakuin University |
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Principal Investigator |
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| Project Period (FY) |
2018-04-01 – 2024-03-31
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| Keywords | 代数曲線束 / 代数曲面 / 分岐被覆 / モジュライ空間 / 複素微分方程式 |
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| Outline of Final Research Achievements |
In this project, we studied the problem of the geography of fibrations of algebraic curves on a non-singular projective algebraic curve with specific structures which is the question of what values of the relative Euler-Poincare characteristic, the self-intersection number of the relative canonical divisor and the genus of a fiber can take. We mainly considered fibrations given by triple coverings of projective line bundles, and proved inequalities among these invariants that depending on the type of branch loci. In particular, we show that the lower bounds on the slope and the Euler-Poincare characteristic differ from each other depending on whether the genus of a fiber is congruent to 2 modulo 3 or not. Furthermore, we specifically gave triple coverings of projective line bundles and proved the existence of fibrations of algebraic curves.
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| Free Research Field |
代数幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
代数曲線束のジオグラフィーに関する主要な問題の1つである「特定の構造をもつ代数曲線束の不変量がどのような値を取り得るか」に関して，先行研究が豊富とは言えなかった．
本研究課題の成果により, 特にファイバーの種数が3を法として2と合同であるかどうかによって，射影直線束の3重被覆で与えられる代数曲線束の不変量に制約があることが明確になった．他の代数曲線束の構造においても，本成果と同様の現象が予想でき，今後の代数曲線束のジオグラフィーの問題の進展が期待できると考える．
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