Arithmetic Geometry via Higher Dimensional Algebraic Geometry

2021 Fiscal Year Final Research Report

• Project/Area Number: 19K14512
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Nagoya University (2021); Kumamoto University (2019-2020)
• Principal Investigator: Tanimoto Sho 名古屋大学, 多元数理科学研究科, 准教授 (10785786)
• Project Period (FY): 2019-04-01 – 2022-03-31
• Keywords: Manin予想 / Fano多様体 / 有理点 / 曲線 / モジュライ空間 / 曲げ折り法 / Campana点 / 極小モデル理論

Outline of Final Research Achievements

Rational solutions to a system of polynomial equations have been studied since the age of Greece. Polynomial equations define an algebraic variety investigated by algebraic geometers, and rational solutions are called as rational points on the variety. When there are infinitely many rational points on an algebraic variety, one can consider the counting function of rational points on that variety. One of outstanding questions is an asymptotic formula of this counting function, and this asymptotic formula is predicted by Manin's conjecture. In our research, we resolved a long standing mystery of exceptional sets in Manin's conjecture. Moreover we formulated Manin's conjecture for Campana points which are intermediate objects between rational points and integral points. Finally using analogy between rational points and curves, we studied properties of moduli spaces of curves on an algebraic variety using the perspective of Manin's conjecture.

Free Research Field

代数幾何, 数論幾何

Academic Significance and Societal Importance of the Research Achievements

Manin予想の例外集合の双有理幾何学にまつわる研究は, Manin予想の理論の根幹をなす研究といえ, 専門家から高い評価を受けています. 私たちが発表した論文は希薄集合版のManin予想について基本的な文献になりつつあります. さらにCampana点のManin予想に関する研究は, 私たちの論文が発表された以降数多くのCampana点のManin予想に関する研究が生まれました. さらに曲線のモジュライ空間にまつわる研究は, 一つのムーブメントとして専門家から捉えられ, 若い数学者が研究に参画してきています.