Birational Mori fiber structures of Fano varieties and rationality problem

• Project/Area Number: 26800019
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Saga University
• Principal Investigator: Okada Takuzo 佐賀大学, 工学(系)研究科(研究院), 准教授 (20547012)
• Project Period (FY): 2014-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000); Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2014: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: ファノ多様体 / del Pezzo束 / コニック束 / 有理性問題 / 安定有理性 / del Pezzo 束 / 森ファイバー空間

Outline of Final Research Achievements

We determined the birational Mori fiber structures of members of 21 families of 3 dimensional Fano weighted complete intersections of codimension 2. Moreover we classified birationally rigid Fano threefolds embedded as a codimension 3 subvariety in a weighted projective space. As a consequence, the classification of birationally rigid Fano threefolds which are embedded as a codimension at most 3 subvariety in a weighted projective space is completed. We also proved the non-rationality of some 3-dimensional del Pezzo fibrations admitting an action of the Klein simple group, and proved K-stability of various birationally rigid Fano threefolds.

Research Products

• [Int'l Joint Research] Loughborough University(英国)
• [Int'l Joint Research] Seoul National University/Institute for Basic Science(韓国)
• [Int'l Joint Research] Loughborough University(英国)
• [Int'l Joint Research] Max Planck Institute(ドイツ)
• [Int'l Joint Research] Seoul National University/IBS Center for Geometry and Physics(韓国)
• [Journal Article] Birationally rigid Pfaffian Fano 3-folds (2018)
  • Author(s): Hamid Ahmadinezhad, Takuzo Okada
  • Journal Title: Algebraic Geometry Volume: 5 Pages: 160-199
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Birational Mori fiber structures of Q-Fano 3-fold weighted complete intersections, II (2018)
  • Author(s): Takuzo Okada
  • Journal Title: Journal fur die reine und angewandte Mathematik Volume: 印刷中 Issue: 738 Pages: 160-199
  • DOI: 10.1515/crelle-2015-0054
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Alpha Invariants of Birationally Rigid Fano Three-folds (2018)
  • Author(s): In-Kyun Kim, Takuzo Okada, Joonyeong Won
  • Journal Title: International Mathematics Research Notices Volume: 印刷中 Pages: 2745-2800
  • DOI: 10.1093/imrn/rnw310
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Q-Fano threefolds with three birational Mori fiber structures (2017)
  • Author(s): Takuzo Okada
  • Journal Title: "Algebraic Varieties and Automorphism Groups", Advanced Studies in Pure Mathematics Volume: 75 Pages: 393-424
  • Peer Reviewed
• [Journal Article] Nonrational del Pezzo fibrations admitting an action of the Klein simple group (2016)
  • Author(s): Takuzo Okada
  • Journal Title: European Journal of Mathematics Volume: 2 Issue: 1 Pages: 319-332
  • DOI: 10.1007/s40879-015-0080-2
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Birational Mori fiber structures of Q-Fano threefold weighted complete intersections (2014)
  • Author(s): Takuzo Okada
  • Journal Title: Proc. London Math. Soc. Volume: 109 Issue: 6 Pages: 1549-1600
  • DOI: 10.1112/plms/pdu044
  • Peer Reviewed
• [Presentation] On birational rigidity of singular del Pezzo fibrations of degree 1 (2018)
  • Author(s): Takuzo Okada
  • Organizer: The 16th Affine Algebraic Geometry Meeting
  • Int'l Joint Research / Invited
• [Presentation] Smooth weighted hypersurfaces that are not stably rational (2017)
  • Author(s): Takuzo Okada
  • Organizer: Birkar's boundedness and Cremona groups (Edge days 2017)
  • Int'l Joint Research / Invited
• [Presentation] On stable rationality problem of Fano varieties (2017)
  • Author(s): Takuzo Okada
  • Organizer: Stability, Boundedness and Fano varieties
  • Int'l Joint Research / Invited
• [Presentation] On stable rationality of del Pezzo fibrations (2017)
  • Author(s): Takuzo Okada
  • Organizer: アフィン代数幾何学研究集会
  • Place of Presentation: 関西学院大学大阪梅田キャンパス
  • Int'l Joint Research / Invited
• [Presentation] Stable rationality of conic bundles over projective spaces (2017)
  • Author(s): 岡田拓三
  • Organizer: 農工大数学セミナー2017
  • Place of Presentation: 東京農工大学小金井キャンパス
  • Invited
• [Presentation] Stable rationality of higher dimensional conic bundles (2017)
  • Author(s): 岡田拓三
  • Organizer: 野田シンポジウム
  • Place of Presentation: 東京理科大学野田キャンパス
  • Invited
• [Presentation] Non-rationality of del Pezzo fibrations by reduction modulo p (2016)
  • Author(s): Takuzo Okada
  • Organizer: Linfoot Seminar
  • Place of Presentation: University of Bristol, UK
  • Year and Date: 2016-02-17
  • Int'l Joint Research / Invited
• [Presentation] Birationally bi-rigid Fano threefolds of codimension 2 (2016)
  • Author(s): Takuzo Okada
  • Organizer: Heilbronn Focused Research Workshop on Birational Geometry
  • Place of Presentation: University of Warwick, UK
  • Year and Date: 2016-02-11
  • Int'l Joint Research / Invited
• [Presentation] Classification of birationally bi-rigid Fano 3-fold weighted complete intersections (2016)
  • Author(s): Takuzo Okada
  • Organizer: Higher dimensional algebraic geometry and around, Kobe-Kyoto
  • Place of Presentation: 京都大学
  • Year and Date: 2016-02-04
  • Int'l Joint Research / Invited
• [Presentation] Stable rationality of cyclic covers and some orbifold Fano 3-folds (2016)
  • Author(s): Takuzo Okada
  • Organizer: Workshop on birational geometry and reduction to positive characteristic
  • Place of Presentation: University of Edinburgh
  • Int'l Joint Research / Invited
• [Presentation] Stable rationality of orbifold Fano 3-fold hypersurfaces (2016)
  • Author(s): Takuzo Okada
  • Organizer: Rationality and selfmaps
  • Place of Presentation: 京都大学数理解析研究所
  • Int'l Joint Research / Invited
• [Presentation] Stable rationality of orbifold Fano 3-fold hypersurfaces (2016)
  • Author(s): Takuzo Okada
  • Organizer: 城崎代数幾何学シンポジウム
  • Place of Presentation: 城崎国際アートセンター
  • Int'l Joint Research / Invited
• [Presentation] Birationally rigid Pfaffian Fano 3-folds (2015)
  • Author(s): 岡田拓三
  • Organizer: 第3回杜の都代数幾何研究集会
  • Place of Presentation: 福岡大学セミナーハウス
  • Year and Date: 2015-11-26
  • Invited
• [Presentation] 3次元ファノ多様体の双有理幾何に着いて (2015)
  • Author(s): 岡田拓三
  • Organizer: 第133回日本数学会九州支部例会（特別講演）
  • Place of Presentation: 佐賀大学
  • Year and Date: 2015-10-24
  • Invited
• [Presentation] Birational geometry of Fano 3-folds of codimension 2 and 3 (2015)
  • Author(s): Takuzo Okada
  • Organizer: Algebraic Geometry Seminar
  • Place of Presentation: IBS Center for Geometry and Physics, Pohang, Korea
  • Year and Date: 2015-09-15
  • Int'l Joint Research / Invited
• [Presentation] Nonrational del Pezzo fibrations admitting an action of the Klein simple group (2015)
  • Author(s): Takuzo Okada
  • Organizer: Workshop on singular del Pezzo surfaces (Edge days 2015)
  • Place of Presentation: University of Edinburgh, UK
  • Year and Date: 2015-06-05
  • Int'l Joint Research / Invited
• [Presentation] 3次元Qファノ重み付き完全交叉の双有理的森ファイバー構造について (2014)
  • Author(s): 岡田拓三
  • Organizer: 第12回代数曲線論シンポジウム
  • Place of Presentation: 日本大学理工学部
  • Year and Date: 2014-12-20
  • Invited
• [Presentation] Birational geometry of Q-Fano 3-fold weighted complete intersections (2014)
  • Author(s): Takuzo Okada
  • Organizer: Landau-Ginzburg theory and Fano varieties
  • Place of Presentation: Gyeongyu, Korea
  • Year and Date: 2014-05-26
  • Invited
• [Funded Workshop] Younger Generation in Algebraic and Complex Geometry (2016)
  • Place of Presentation: National Taiwan University
  • Year and Date: 2016-01-11