Computational study of K3 surfaces

2015 Fiscal Year Final Research Report

• Project/Area Number: 25400042
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Hiroshima University
• Principal Investigator: Shimada Ichiro 広島大学, 理学(系)研究科(研究院), 教授 (10235616)
• Co-Investigator(Kenkyū-buntansha): Ishii Akira 広島大学, 大学院理学研究科, 教授 (10252420) ; Kimura Shun-ichi 広島大学, 大学院理学研究科, 教授 (10284150) ; Takahashi Nobuyoshi 広島大学, 大学院理学研究科, 准教授 (60301298) ; Matsumoto Makoto 広島大学, 大学院理学研究科, 教授 (70231602) ; Hiranouchi Toshiro 広島大学, 大学院理学研究科, 助教 (30532551) ; Takahashi Hiroki 徳島大学, ソシオテクノサイエンス研究部, 教授 (90291476)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Keywords: K3曲面 / 計算機アルゴリズム / 格子 / 自己同型 / 射影モデル

Outline of Final Research Achievements

By means of computer-aided calculations of lattices, we obtained many geometric consequences on K3 surfaces and related algebraic varieties.　(1) We determined finite sets of generators of automorphism groups of several singular K3 surfaces by generalized Borcherds-Kondo method. (2) We obtained three interesting projective models of the supersingular K3 surface with Artin invariant 1 in characteristic 5 (joint work with T.Katsura and S.Kondo). This result led us to a new construction of the Hoffman-Singleton graph and the Higman-Sims graph. (3) By an experimental computation, we presented a conjecture that each supersingular K3 surface in odd characteristic has an automorphism whose characteristic polynomial on the Neron-Severi lattice is an irreducible Salem polynomial. (4) We presented a combinatorial description of the submodule of the middle homology group of an even dimensional complex Fermat variety generated by the classes of linear subspaces (joint work with A. Degtyarev).

Free Research Field

代数幾何学