Noncommutative algebraic geometry

2023 Fiscal Year Final Research Report

• Project/Area Number: 19KK0348
• Research Category: Fund for the Promotion of Joint International Research (Fostering Joint International Research (A))
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Osaka University
• Principal Investigator: Okawa Shinnosuke 大阪大学, 大学院理学研究科, 准教授 (60646909)
• Project Period (FY): 2021 – 2023
• Keywords: 非可換代数幾何学 / del Pezzo曲面 / ワイル群

Outline of Final Research Achievements

The algebraic surface obtained by blowing up the projective plane in 6 points in a general position is isomorphic to a cubic surface in a 3-dimensional projective space. Conversely, all (nonsingular) cubic surfaces are always constructed in this way. Moreover there is arbitrariness in the configuration of the 6 points which yield the given cubic surface, which coincides exactly with the orbit of an action of the Weyl group of type E6. In this joint work we generalize this monumental classical result of algebraic geometry from the 19th century to noncommutative algebraic varieties. Namely, we succeeded in describing a cubic surface in a given 3-dimensional noncommutative projective space as a 6-point blowup of a noncommutative projective plane. This result is achieved by identifying an ``ambient space'' mapping from the moduli space classifying the latter to the moduli space classifying the former. We also elucidated the Poisson geometry obtained as a semi-classical limit.

Free Research Field

非可換代数幾何学

Academic Significance and Societal Importance of the Research Achievements

本研究課題は2000年頃には認識されていた、非可換射影幾何学の重要な問題の一つであった。今回、モジュライ空間の間の有理射を同定するという形で完全解決できた。主結果の定式化の方法、また、直線のHilbert schemeを利用した証明の手法共に、画期的であった。さらに、この結果により、他の次数の非可換del Pezzo曲面の反標準線型系の幾何学を研究する筋道も立った。加えて、非可換代数多様体を調べるうえで対応するPoisson幾何学に注目することの重要性が明らかになったという点も、手法面において重要な発見であったと考えている。