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A study of general members of linear systems via the theory of F-singularities

Research Project

Project/Area Number 17H02831
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionThe University of Tokyo

Principal Investigator

Takagi Shunsuke  東京大学, 大学院数理科学研究科, 教授 (40380670)

Co-Investigator(Kenkyū-buntansha) 權業 善範  東京大学, 大学院数理科学研究科, 准教授 (70634210)
中村 勇哉  東京大学, 大学院数理科学研究科, 助教 (20780034)
Project Period (FY) 2017-04-01 – 2022-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥14,950,000 (Direct Cost: ¥11,500,000、Indirect Cost: ¥3,450,000)
Fiscal Year 2021: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000)
Fiscal Year 2020: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000)
Fiscal Year 2019: ¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000)
Fiscal Year 2018: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Fiscal Year 2017: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
Keywords特異点 / F特異点 / 正標数 / 可換環論 / 極小モデル理論 / ファノ多様体 / 変形 / 特異点の変形 / 付置論的対数端末特異点 / 付置論的対数標準特異点 / Q-Gorenstein / 代数幾何学 / 対数標準的特異点 / F純特異点 / F正則特異点 / 対数標準特異点 / 対数端末特異点 / 大域的F分裂多様体 / 秋月・中野の消滅定理 / 大域的F正則型多様体 / 消滅定理 / 判定イデアル / 数値的Q-Gorenstein / 単射性定理 / 大域的F正則多様体 / 一般元
Outline of Final Research Achievements

In this research project, we mainly studied the deformations of algebraic varieties and singularities defined over a field of positive characteristic. We have two main results. (i) We proved vanishing theorems on globally F-split and globally F-regular varieties and used them to study the flat deformation of a globally F-split Fano 3-fold. (ii) We studied the deformations of F-pure and F-regular singularities. We proved if there exists a single prime number p such that the modulo p reduction of a Q-Gorenstein normal singularity (X, x) is F-pure, then (X,x) is log canonical.

Academic Significance and Societal Importance of the Research Achievements

ホッジ理論などの解析的手法が使えないことから、正標数のファノ多様体の変形についてはほとんど何もわかっていなかった.大域F分裂多様体の枠組みで問題を考えることで,標数0の結果を含意するような正標数の結果を得ることに成功した.
対数標準特異点は極小モデル理論に現れる特異点の中で最も広いクラスであるが,特異点解消を用いて定義されるため,与えられた特異点が対数標準特異点かどうか判定することは難しかった.しかし,本研究成果によって,計算ソフトウェアを使って確認できる十分条件が得られた.

Report

(6 results)
  • 2022 Final Research Report ( PDF )
  • 2021 Annual Research Report
  • 2020 Annual Research Report
  • 2019 Annual Research Report
  • 2018 Annual Research Report
  • 2017 Annual Research Report
  • Research Products

    (41 results)

All 2023 2022 2021 2020 2019 2018 2017 Other

All Int'l Joint Research (4 results) Journal Article (11 results) (of which Int'l Joint Research: 3 results,  Peer Reviewed: 11 results,  Open Access: 1 results) Presentation (20 results) (of which Int'l Joint Research: 14 results,  Invited: 18 results) Remarks (4 results) Funded Workshop (2 results)

  • [Int'l Joint Research] Imperial College London(英国)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] University of Utah(米国)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] FPT University(ベトナム)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] Imperial College London(英国)

    • Related Report
      2017 Annual Research Report
  • [Journal Article] Deformations of log terminal and semi log canonical singularities2023

    • Author(s)
      Kenta Sato and Shunsuke Takagi
    • Journal Title

      Forum of Mathematics, Sigma

      Volume: 11 Pages: 1-34

    • DOI

      10.1017/fms.2023.28

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] A Witt Nadel vanishing theorem for threefolds2020

    • Author(s)
      Yusuke Nakamura, Hiromu Tanaka
    • Journal Title

      Compos. Math.

      Volume: 156 Issue: 3 Pages: 435-475

    • DOI

      10.1112/s0010437x1900770x

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Minimal model program for log canonical threefolds in positive characteristic2020

    • Author(s)
      Kenta Hashizume, Yusuke Nakamura, Hiromu Tanaka
    • Journal Title

      Mathematical Research Letters

      Volume: -

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Rational points on log Fano threefolds over a finite field2019

    • Author(s)
      Yoshinori Gongyo, Yusuke Nakamura, Hiromu Tanaka
    • Journal Title

      J. Eur. Math. Soc. (JEMS)

      Volume: 21 Issue: 12 Pages: 3759-3795

    • DOI

      10.4171/jems/913

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] General hyperplane sections of threefolds in positive characteristic2018

    • Author(s)
      Kenta Sato and Shunsuke Takagi
    • Journal Title

      Journal of the Institute of Mathematics of Jussieu

      Volume: 印刷中 Issue: 2 Pages: 647-661

    • DOI

      10.1017/s1474748018000166

    • Related Report
      2019 Annual Research Report 2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Koll\'ar’s injectivity theorem for globally F-regular varieties2018

    • Author(s)
      Gongyo Yoshinori、Takagi Shunsuke
    • Journal Title

      European Journal of Mathematics

      Volume: 印刷中 Issue: 3 Pages: 872-880

    • DOI

      10.1007/s40879-018-0230-4

    • Related Report
      2019 Annual Research Report 2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Finitistic test ideals on numerically Q-Gorenstein varieties2018

    • Author(s)
      Shunsuke Takagi
    • Journal Title

      J. Algebra

      Volume: 印刷中 Pages: 266-279

    • DOI

      10.1016/j.jalgebra.2018.08.003

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A boundedness conjecture for minimal log discrepancies on a fixed germ2018

    • Author(s)
      Mircea Mustata, Yusuke Nakamura
    • Journal Title

      Contemporary Mathematics

      Volume: 712 Pages: 287-306

    • DOI

      10.1090/conm/712/14351

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] F-singularities: applications of characteristic p methods to singularity theory2018

    • Author(s)
      Shunsuke Takagi and Kei-ichi Watanabe
    • Journal Title

      Sugaku Expositions

      Volume: 31 Issue: 1 Pages: 1-42

    • DOI

      10.1090/suga/427

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A boundedness conjecture for minimal log discrepancies on a fixed germ2018

    • Author(s)
      M. Mustata and Yusuke Nakamura
    • Journal Title

      Contemporary Mathematics

      Volume: 印刷中

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] A Gorenstein criterion for strongly F-regular and log terminal singularities2017

    • Author(s)
      Anurag Singh, Shunsuke Takagi and Matteo Varbaro
    • Journal Title

      International Mathematics Research Notices

      Volume: 2017 Pages: 6484-6522

    • DOI

      10.1093/imrn/rnw290

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Pure subrings of KLT singularities and BCM test ideals2023

    • Author(s)
      Shunsuke Takagi
    • Organizer
      Resolutions in Local Algebra and Singularity Theory
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 正則有限被覆を持つ特異点の因子類群について2022

    • Author(s)
      高木 俊輔
    • Organizer
      可換環論の新しい融合セミナー
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] 対数端末特異点と対数標準特異点の変形について2022

    • Author(s)
      高木 俊輔
    • Organizer
      名古屋大学代数幾何学セミナー
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] Kodaira vanishing for thickenings of globally F-regular varieties2021

    • Author(s)
      Shunsuke Takagi
    • Organizer
      IIT Bombay Virtual Commutative Algebra Seminar
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Arithmetic deformations of F-singularities2021

    • Author(s)
      Shunsuke Takagi
    • Organizer
      MSRI Fellowship of the Ring, National Seminar
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Arithmetic deformations of F-pure singularities2021

    • Author(s)
      Shunsuke Takagi
    • Organizer
      OIST Workshop ”Quantum Math, Singularities and Applications”
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Deformations of F-pure and F-regular singularities2021

    • Author(s)
      Shunsuke Takagi
    • Organizer
      Zoom Algebraic Geometry Seminar
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On vanishing theorems for Frobenius split varieties in positive characteristic2020

    • Author(s)
      Shunsuke Takagi
    • Organizer
      Joint Tokyo/Kyoto Algebraic Geometry Seminar
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] F-singularities and singularities in birational geometry I, II2020

    • Author(s)
      Shunsuke Takagi
    • Organizer
      IIT Bombay Virtual Commutative Algebra Seminar
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Vanishing theorems on globally F-regular varieties2019

    • Author(s)
      Shunsuke Takagi
    • Organizer
      Special Conference on Commutative Algebra and its Interaction with Algebraic Geometry
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Kodaira vanishing for thickenings of globally F-regular threefolds2019

    • Author(s)
      Shunsuke Takagi
    • Organizer
      第41回可換環論シンポジウム
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Weak Akizuki-Nakano vanishing for globally F-split 3-folds2019

    • Author(s)
      Shunsuke Takagi
    • Organizer
      Tokyo-Seoul Conference in Mathematics 2019 -Algebraic Geometry-
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On threefolds of globally F-regular type2019

    • Author(s)
      Shunsuke Takagi
    • Organizer
      AMS Spring Central and Western Sectional Meeting, Special Session on Commutative Algebra and its Environs
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Test ideals on numerically Q-Gorenstein rings2019

    • Author(s)
      Shunsuke Takagi
    • Organizer
      Singularities and Homological Aspects of Commutative Algebra
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 代数多様体の特異点論と正標数の手法2019

    • Author(s)
      高木 俊輔
    • Organizer
      日本数学会2019年度年会(招待講演)
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Vanishing theorems on globally F-regular varieties2018

    • Author(s)
      Shunsuke Takagi
    • Organizer
      Positivity in Algebraic Geometry
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] General hyperplane sections of threefolds in positive characteristic2018

    • Author(s)
      高木 俊輔
    • Organizer
      第4回杜の都代数幾何学研究集会
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] Finitistic test ideals on numerically Q-Gorenstein varieties2018

    • Author(s)
      高木 俊輔
    • Organizer
      第30回可換環論セミナー
    • Related Report
      2017 Annual Research Report
  • [Presentation] 正標数の可換環論におけるフロべニウス射を用いた手法2017

    • Author(s)
      高木 俊輔
    • Organizer
      可換環論と数論幾何の新展開~ホモロジカル予想を通じて~
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] On F-pure thresholds2017

    • Author(s)
      Shunsuke Takagi
    • Organizer
      Commutative Algebra and Related Topics
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Remarks] Shunsuke Takagi

    • URL

      https://www.ms.u-tokyo.ac.jp/~stakagi/

    • Related Report
      2021 Annual Research Report 2020 Annual Research Report 2018 Annual Research Report
  • [Remarks] Shunsuke Takagi

    • URL

      http://www.ms.u-tokyo.ac.jp/~stakagi/

    • Related Report
      2019 Annual Research Report 2017 Annual Research Report
  • [Remarks] 權業 善範(Yoshinori Gongyo)

    • URL

      https://www.ms.u-tokyo.ac.jp/~gongyo/

    • Related Report
      2019 Annual Research Report 2017 Annual Research Report
  • [Remarks] Yusuke Nakamura

    • URL

      https://sites.google.com/site/ynakamuraagmath/

    • Related Report
      2019 Annual Research Report 2017 Annual Research Report
  • [Funded Workshop] Algebraic Geometry in Positive Characteristic and Related Topics2018

    • Related Report
      2018 Annual Research Report
  • [Funded Workshop] 第39回可換環論シンポジウム2017

    • Related Report
      2017 Annual Research Report

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Published: 2017-04-28   Modified: 2024-01-30  

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