Studies on K3 surfaces and related varieties using reduction modulo p

2022 Fiscal Year Final Research Report

• Project/Area Number: 16K17560
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Tokyo University of Science (2018-2022); Nagoya University (2016-2017)
• Principal Investigator: Matsumoto Yuya 東京理科大学, 理工学部数学科, 助教 (50773628)
• Project Period (FY): 2016-04-01 – 2023-03-31
• Keywords: K3曲面 / 正標数 / 有理二重点 / 自己同型 / 混標数 / 群スキーム / Enriques曲面 / Kummer曲面

Outline of Final Research Achievements

We first studied relations between good reduction of K3 surfaces and automorphism groups. Then we focused on actions of finite group schemes μ_p and α_p on K3 surfaces in characteristic p. In characteristic 0, a criterion for the quotient by a finite group being K3 or not was known. We showed an analogue of this criterion in the case of μ_p-action in characteristic p. We also proved a relation with the height of the K3 surfaces, which is an invariant in positive characteristic.
We also studied rational double point singularities in positive characteristic. We showed that some of them are quotients by finite group schemes and that some are not.

Free Research Field

代数幾何学

Academic Significance and Societal Importance of the Research Achievements

標数pのK3曲面へのμ_pやα_pといった有限群スキームの作用に豊富な理論が存在することを見出したことで，正標数のK3曲面や関連する代数多様体の研究に新たな展開をもたらすと考える．また，正標数の特異点について有限群だけでなく有限群スキームの商として考える視点を与え，それらの性質を調べたことで，正標数の特異点の理論に貢献するものと考える．