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Noncommutative algebraic geometry

Research Project

Project/Area Number 19KK0348
Research Category

Fund for the Promotion of Joint International Research (Fostering Joint International Research (A))

Allocation TypeMulti-year Fund
Research Field Algebra
Research InstitutionOsaka 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
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥11,830,000 (Direct Cost: ¥9,100,000、Indirect Cost: ¥2,730,000)
Keywords非可換代数幾何学 / del Pezzo曲面 / ワイル群 / 3次曲面 / 導来圏
Outline of Research at the Start

Del Pezzo曲面は最もよく研究されてきた代数曲面の1種である。それらの一般化である非可換del Pezzo曲面は様々な分野と関わる大変興味深い研究対象である。
Del Pezzo曲面は2次元のFano多様体のことであり、その反標準線型系の幾何学が重要である。例えば3次のdel Pezzo曲面は1本の3次方程式で定義される曲面に他ならないが、この大事な事実は反標準線型系の幾何学で説明される。
本研究は、これを非可換del Pezzo曲面に拡張するものである。

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.

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。

Report

(4 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • Research Products

    (11 results)

All 2023 2021 Other

All Int'l Joint Research (2 results) Presentation (7 results) (of which Int'l Joint Research: 5 results,  Invited: 7 results) Remarks (2 results)

  • [Int'l Joint Research] Hasselt 大学(ベルギー)2021

    • Year and Date
      2021-10-01
    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] ブリュッセル自由大学(VUB)(ベルギー)2021

    • Year and Date
      2021-10-01
    • Related Report
      2023 Annual Research Report
  • [Presentation] Blowing down noncommutative cubic surfaces2023

    • Author(s)
      Shinnosuke Okawa
    • Organizer
      NCTS Higher Dimensional Algebraic Geometry Mini-courses and Workshop
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Blowing down noncommutative cubic surfaces2023

    • Author(s)
      大川新之介
    • Organizer
      杉本代数セミナー
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Blowing down noncommutative cubic surfaces2023

    • Author(s)
      大川新之介
    • Organizer
      新潟代数シンポジウム
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Blowing down noncommutative cubic surfaces2023

    • Author(s)
      Shinnosuke Okawa
    • Organizer
      2023 NCTS Higher Dimensional Algebraic Geometry Mini-courses and Workshop
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Semiorthogonal indecomposability of irregular surfaces2023

    • Author(s)
      Shinnosuke Okawa
    • Organizer
      The 1st Algebraic geometry Atami symposium
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Blowing down noncommutative cubic surfaces2023

    • Author(s)
      Shinnosuke Okawa
    • Organizer
      CURRENT TRENDS IN THE CATEGORICAL APPROACH TO ALGEBRAIC AND SYMPLECTIC GEOMETRY
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The reconstruction theorem for AS-regular 3-dimensional cubic Z-algebras2023

    • Author(s)
      Shinnosuke Okawa
    • Organizer
      Interactions between Algebraic Geometry and Noncommutative Algebra (Oberwolfach Workshop 2218)
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Remarks] arXiv:2304.14048

    • URL

      https://arxiv.org/abs/2304.14048

    • Related Report
      2022 Research-status Report
  • [Remarks] arXiv:2206.13359

    • URL

      https://arxiv.org/abs/2206.13359

    • Related Report
      2022 Research-status Report

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Published: 2020-02-06   Modified: 2025-01-30  

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