Deeping algebraic geometry in positive characteristic

2022 Fiscal Year Final Research Report

• Project/Area Number: 16K05109
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Tokyo University of Science
• Principal Investigator: Hiroyuki Ito 東京理科大学, 理工学部数学科, 教授 (60232469)
• Project Period (FY): 2016-04-01 – 2023-03-31
• Keywords: 正標数代数幾何学 / 特異点 / K3曲面

Outline of Final Research Achievements

We have obtained the following great results on 2-dimensional quotient singularities in positive characteristic.
By defining a pseudo-derivation that generalizes the derivation corresponding to the length-p finite group scheme of additive type, we connect this group scheme action with the Artin-Schreier type group action, that is, the wild cyclic group action by deformation. That makes a theory for unified p-group schemes. This leads to a background understanding that the rational double point is not a Taut singularity, especially in the case of low characteristic, and it seems to have been the first step toward understanding the McKay correspondence.
In addition, we describe a strata with Artin invariant 3 on a supersingular K3 surface in characteristic 2 using quasi-elliptic fibrations, and at the same time we obtain interesting linear signatures.

Free Research Field

代数幾何学

Academic Significance and Societal Importance of the Research Achievements

正標数代数多様体の理解は代数幾何学や数論幾何学において非常に重要である。特に正標数をも包含した一般理論構築は代数多様体の理解に不可欠である。また、特異点理論についても同様である。代数幾何学や特異点論における一般理論構築の障害として低標数の例外的な現象がある。本研究はその様な例外的現象をも包含する背景理解や一般理論構築を目指すものであり、得られた研究成果は確実に代数幾何学や数論幾何学の発展に寄与するものと考えられる。