Studies of K3 surfaces and rational surfaces through symmetry

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K03454
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Tokai University
• Principal Investigator: Taki Shingo 東海大学, 理学部, 准教授 (30609714)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: K3曲面 / 自己同型 / 商曲面 / Galois点

Outline of Final Research Achievements

It is a fundamental problem to study automorphisms of algebraic varieties. In particular, studies on automorphisms of K3 surfaces are one of the important problems. By the definition of K3 surfaces, these have a nowhere vanishing holomorphic 2-form. A finite group which acts on K3 surfaces as automorphisms is called "symplectic" or "non-symplectic" if it acts trivially or non-trivially on a nowhere vanishing holomorphic 2-form, respectively.
In this research project, study finite non-symplectic automorphisms on K3 surfaces from the following three view points.
(1) A study on quotient surfaces of K3 surfaces with an automorphism of maximum order. (2) A study on non-purely non-symplectic automorphisms on K3 surfaces. (3) A study on characterization of quartic surfaces with Galois points as K3 surface

Free Research Field

代数幾何

Academic Significance and Societal Importance of the Research Achievements

幾何的対象を考察する際，それが持つ対称性がその特殊性を表していることがある．例えば，一般の三角形に対称性は無いが，二等辺三角形や正三角形のような特殊な三角形は「左右対称」や「120度の回転」など対称性を持つ．
本研究ではこのような視点の下，K3曲面とよばれる代数多様体の対称性を調べ，「K3曲面上のある種の特別な現象の背景には特殊な対称性が隠されている」ということを示した．