Combinatorial structure on algebraic varieties in positive characteristics

2018 Fiscal Year Final Research Report

• Project/Area Number: 16K13749
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Hiroshima University
• Principal Investigator: Shimada Ichiro 広島大学, 理学研究科, 教授 (10235616)
• Co-Investigator(Kenkyū-buntansha): 木村 俊一 広島大学, 理学研究科, 教授 (10284150) ; 奥田 隆幸 広島大学, 理学研究科, 講師 (40725131)
• Project Period (FY): 2016-04-01 – 2019-03-31
• Keywords: 格子 / 符号 / 計算機アルゴリズム

Outline of Final Research Achievements

We investigated various algebraic varieties from combinatorial point of view, and observed many interesting facts. During the investigation, we improved several algorithms for combinatorial calculations, and implemented them in our computers. Thanks to these algorithms, we have succeeded in probing geometric properties of some K3 and Enriques surfaces, and in particular, determined their automorphism groups.
We also made huge data about the Leech lattice. We hope this data will be helpful in future investigation of algebraic varieties from combinatorial point of view.

Free Research Field

代数幾何学

Academic Significance and Societal Importance of the Research Achievements

格子や符号といった組み合わせ論的な数学的対象は，安全で高速なデジタル通信を支える上で重要な役割を果たしており，良い格子，良い符号，良い組み合わせ論的対象を見つけることは数学の重要な課題の一つである．特に，対称性の高い特別な代数多様体の幾何学からは興味深い組み合わせ論的構造が得られることが多い．本研究においては，様々な代数多様体に現れる興味深い組み合わせ論的構造を調べた．