Study on Fano varieties defined over an algebraically closed field in positive characteristic

2021 Fiscal Year Final Research Report

• Project/Area Number: 17K05208
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Hiroshima City University
• Principal Investigator: Saito Natsuo 広島市立大学, 情報科学研究科, 准教授 (70382372)
• Project Period (FY): 2017-04-01 – 2022-03-31
• Keywords: 正標数 / del Pezzo曲面 / 準楕円曲面

Outline of Final Research Achievements

We studied various properties about Fano varieties over an algebraic closed field in positive characteristic to have the following result:
1. We investigated non-F-split del Pezzo surfaces of degree 1. Especially, we showed that such surfaces are unique up to isomorphisms if the characteristic of the grould field is 5, and its automorphism group is related to the fact that the symmetric group of degree 6 has an outer automorphism.
2. We solved an open problem on the multicanonical system of quasi-elliptic surfaces with Kodaira dimension 1 in characteristic 2. Also, we analyzed the structure of Mordell-Weil group of quasi-elliptic K3 surfaces which has twenty reducible fibers of type III in characteristic 2, to construct a 20-dimensional linear code.

Free Research Field

代数幾何学

Academic Significance and Societal Importance of the Research Achievements

代数多様体の分類を完成させることは代数幾何学における大きな研究テーマであるが，正標数の体上では分類に役立つ定理のいくつかが成立せず，理論の構築は容易ではない。したがって，分類を行ううえで重要な役割を果たすFano多様体やそれにまつわる正標数特有の幾何的構造を解明することは大きな意義がある。本研究において特に低標数の場合に発生するいくつかの特殊な構造を記述することに成功したことは，正標数の代数多様体の分類理論の発展に資するものである。