The solution of Hurwitz's problem through Galois covers of algebraic curves and study on curves on K3 surfaces
| Project/Area Number |
18K03228
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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 | Kanagawa Institute of Technology |
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Principal Investigator |
Komeda Jiryo 神奈川工科大学, 公私立大学の部局等, 教授 (90162065)
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| Project Period (FY) |
2018-04-01 – 2022-03-31
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| Project Status |
Completed (Fiscal Year 2021)
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| Budget Amount *help |
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2020: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2019: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2018: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
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| Keywords | 代数曲線 / ワイエルシュトラス半群 / トーリック曲面 / K３曲面 / ２重被覆 / ３重被覆 / ガロア直線 / 数値半群 / Almost symmetric 数値半群 / 二重被覆 / 代数曲線の三重被覆 / 有理楕円曲面 / Weierstrass semigroups / Numerical semigroups / Non-singular curves / K3 surfaces / Toric surfaces / Cyclic covers of curves / Triple covers of curves / Galois varieties / 三重被覆 / 平面代数曲線 / ガロア被覆 / シグマ関数 |
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| Outline of Final Research Achievements |
What kinds of numerical semigroups are Weierstrass? Namely, what is the condition for a numerical semigroup to be attained by a pointed algebraic curves? This problem is called Hurwitz' Problem. We studied on this problem through the ramification points of double or triple coverings. Especially, in the case where the conductor of a numerical semigroup is fixed we constructed infinite sequences of non-Weierstrass numerical semigroups, i.e., numerical semigroups which cannot be attained by any pointed curves. We studied on pointed algebraic curves on algebraic surfaces. The research objects of the surfaces are the projective plane, toric surfaces and K3 surfaces. The Weierstrass semigroups of pointed algebraic curves on these surfaces were calculated and characterized. Moreover, we gave examples of Weierstrass numerical semigroups which cannot be attained by any pointed curves on these surfaces. Especially, the examples related to toric surfaces are the first ones.
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| Academic Significance and Societal Importance of the Research Achievements |
代数曲線（１次元）を調べるために次元を下げて、その上の点（０次元）を調べる。そのためには、点についての情報が必要になり、それが点のワイエルシュトラス半群である。どのようなワイエルシュトラス半群を持つかで代数曲線を特徴づける。また、ワイエルシュトラス半群を点の性質を忘れて拡張した概念が数値半群である。数値半群がワイエルシュトラス半群になることの必要十分条件を見つけることで、１次元の幾何学的性質を特徴づけることができる。これらのことに関して完全に解決はしていないが、多くの研究成果は得ている。
さらに、いくつかの代数曲面（２次元）を調べるためにその上の１点付き代数曲線（１次元）も調べている。
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Report
(5 results)
Research Products
(27 results)
