2006 Fiscal Year Final Research Report Summary
Study on Galois embeddings of algebraic surfaces
| Project/Area Number |
17540018
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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 |
Algebra
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| Research Institution | Niigata University |
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Principal Investigator |
YOSHIHARA Hisao Niigata University, Institute of Science and Technology, Professor, 自然科学系, 教授 (60114807)
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| Co-Investigator(Kenkyū-buntansha) |
OHBUCHI Akira Tokushima University, Faculty of integrated arts and sciences, Professor, 総合科学部, 教授 (10211111)
KONNO Kazuhiro Osaka University, Graduate school of science, Professor, 大学院理学研究科, 教授 (10186869)
TOKUNAGA Hiro-o Tokyo metropolitan University, Department of mathematics and informatics, Professor, 理工学研究科, 教授 (30211395)
TAKATA Toshie Niigata University, Institute of Science and Technology, Assistant Professor, 自然科学系, 助教授 (40253398)
KOJIMA Hideo Niigata University, Institute of Science and Technology, Assistant Professor, 自然科学系, 助教授 (90332824)
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| Project Period (FY) |
2005 – 2006
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| Keywords | algebraic surface / algebraic curve / Galois point / Galois embedding / abelian surface / birational transformation / Galois group |
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| Research Abstract |
We have studied the Galois embedding of algebraic curves and surfaces, especially rational curves and abelian surfaces.
In the case of abelian surfaces we have obtained all the possible types of the Galois groups which can appear as the covering transformation groups. Moreove we listed a lot of examples of abelian surfaces with given Galois groups of embeddings. In particular we have shown that such abelian surfaces are isogenous to the products of two elliptic curves. On the other hand, we have found the least number N such that abelinan surfaces have the embeddings into PAN. Concernig this study we have studied for singular plane rational curves. We determined all possible type of Galois group, i.e., they are cyclic, dihedral, A_4, S_4 and S_4 and have shown the examples with such Galois groups. Connecting with this research, we have studied if the Galois automorphism can be extended to a birational transformation or not. As a result we have obtained that there are a lot of rational curves such that the autopmorphism cannot be extended to birational transformation, for example we found rational curve with only nodes as singularities and the degree is bigger than 7 with outer Galois point.
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