2020 Fiscal Year Final Research Report
Arithmetic of algebraic varieties with trivial canonical bundle
| Project/Area Number |
15H03614
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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 | The University of Tokyo (2019-2020)
Hosei University (2015-2018) |
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Principal Investigator |
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| Project Period (FY) |
2015-04-01 – 2020-03-31
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| Keywords | 代数曲面 / 標準因子 / Enriques曲面 / 準楕円曲面 / 自己同型群 / 正標数 / K3曲面 / nordal曲線 |
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| Outline of Final Research Achievements |
Enriques surfaces were found at the end of 19 century, and they are important surfaces which belong to one class in the classification theory of algebraic surfaces. When their automorphism groups are finite, they were classified into 7 types over the complex number field by S. Kondo. G. Martin got a similar classification in characteristic p > 2. In our research, we studied Enriques surfaces with finite automorphism group in characteristic 2, and we showed that in the case of classical Enriques surfaces they are classified into 8 types and in the case of supersingular Enriques surfaces they are classified into 5 types. We also got some new results on the K3 surfaces with 9 cusp singularities, on the Zariski property of supersingular K3 surfaces, and on the fiber structures of the quasi-elliptic surfaces with Kodaira dimension 1.
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| Free Research Field |
数学（代数幾何学）
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| Academic Significance and Societal Importance of the Research Achievements |
代数曲面の組織的な研究は19 世紀末のイタリア学派の研究に始まり、複素数体上は小平邦彦による詳細な研究によってその分類を含む理論が構築された。1977年、BombieriとMumfordは正標数の代数的閉体上の代数曲面の分類理論を完成させ、標数2のEnriques曲面や準楕円曲面が明確に認識されるようになった。本研究はそれらの研究に基づき、有限自己同型をもつEnriques曲面の分類に決着をつけたものである。もう一つの研究対象となったK3曲面は、素粒子論で最近用いられている狭義のCalabi-Yau多様体の次元が最も小さい場合であるが、正標数においてその性質のいくつかを解明した。
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