Birational geometry of nonrational rationally connected varieties

• Project/Area Number: 21840032
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Single-year Grants
• Research Field: Algebra
• Research Institution: Kyoto University
• Principal Investigator: OKADA Takuzo Kyoto University, 数理解析研究所, 特定研究員(グローバルCOE) (20547012)
• Project Period (FY): 2009 – 2010
• Project Status: Completed (Fiscal Year 2010)
• Budget Amount: ¥2,262,000 (Direct Cost: ¥1,740,000、Indirect Cost: ¥522,000); Fiscal Year 2010: ¥936,000 (Direct Cost: ¥720,000、Indirect Cost: ¥216,000); Fiscal Year 2009: ¥1,326,000 (Direct Cost: ¥1,020,000、Indirect Cost: ¥306,000)
• Keywords: 有理連結多様体 / ファノ多様体 / 双有理的非有界性 / Qファノ多様体 / 森ファイバー空間 / 双有理的有界性 / 重み付き超曲面

Research Abstract

Q-Fano varieties appear as one of the final outcomes of the (log) minimal model program and they are important objects in the classification theory of algebraic varieties. We studied the birational unboundedness of higher-dimensional Q-Fano varieties. The birational unboundedness means that varieties in concern form infinitely many families even if we identify varieties which are birational to each other. We have proved the birational unboundedness of Q-Fano varieties in each dimension at least 3. We further proved a similar result for rationally connected Mori fiber spaces.

Research Products

• [Journal Article] Birational unboundedness of log terminal Q-Fano varieties and rationally connected strict Mori fiber spaces (2010)
  • Author(s): Takuzo Okada
  • Journal Title: RIMS preprint RIMS-1706 Pages: 1-27
• [Journal Article] Birational unboundedness of log terminal Q-Fano varieties and rationally connected strict Mori fiber spaces (2010)
  • Author(s): Takuzo Okada
  • Journal Title: RIMS Preprint Volume: RIMS-1706 Pages: 1-27
• [Journal Article] On the birational unboundedness of higher dimensional Q-Fano varieties (2009)
  • Author(s): Takuzo Okada
  • Journal Title: Mathematische Annalen Vol.345 Pages: 195-212
  • NAID: 120001723784
  • Peer Reviewed
• [Journal Article] Nonrational weighted hypersurfaces (2009)
  • Author(s): Takuzo Okada
  • Journal Title: Nagoya Mathematical Journal Vol. 194 Pages: 1-32
  • NAID: 120005763694
  • Peer Reviewed
• [Journal Article] On the birational unboundedness of higher dimensional Q-Fano varieties (2009)
  • Author(s): Takuzo Okada
  • Journal Title: Mathematische Annalen Vol. 345 Pages: 195-212
  • NAID: 120001723784
  • Peer Reviewed
• [Presentation] Birational unboundedness of rationally connected Mori fiber spaces (2010)
  • Author(s): 岡田拓三
  • Organizer: 都の西北 代数幾何学シンポジウム
  • Place of Presentation: 早稲田大学
  • Year and Date: 2010-11-12
• [Presentation] Birational unboundedness of rationally connected Mori fiber spaces (2010)
  • Author(s): 岡田拓三
  • Organizer: 都の西北 代数幾何学シンポジウム
  • Place of Presentation: 早稲田大学(招待講演)
  • Year and Date: 2010-11-12
• [Presentation] Birational unboundedness of Q-Fano varieties (2010)
  • Author(s): 岡田拓三
  • Organizer: School (and workshop) on the minimal model program and Shokurov's ACC conjecture
  • Place of Presentation: University of Trento(イタリア)
  • Year and Date: 2010-07-09
• [Presentation] Qファノ多様体の双有理的非有界性 (2010)
  • Author(s): 岡田拓三
  • Organizer: 複素幾何セミナー
  • Place of Presentation: 首都大学東京
  • Year and Date: 2010-05-26
• [Presentation] Qファノ多様体の双有理的非有界性 (2010)
  • Author(s): 岡田拓三
  • Organizer: 複素幾何セミナー
  • Place of Presentation: 首都大学東京(招待講演)
  • Year and Date: 2010-05-26
• [Presentation] Qファノ多様体の双有理的非有界性 (2010)
  • Author(s): 岡田拓三
  • Organizer: 早大理工代数幾何学セミナー
  • Place of Presentation: 早稲田大学
  • Year and Date: 2010-05-21
• [Presentation] Qファノ多様体の双有理的非有界性 (2010)
  • Author(s): 岡田拓三
  • Organizer: 早大理工代数幾何学セミナー
  • Place of Presentation: 早稲田大学(招待講演)
  • Year and Date: 2010-05-21
• [Presentation] Birational unboundedness of Q-Fano varieties of dimension at least five (2010)
  • Author(s): 岡田拓三
  • 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
  • Year and Date: 2010-03-04
• [Presentation] Birational unboundedness of Q-Fano varieties of dimension at least five (2010)
  • Author(s): 岡田拓三
  • Organizer: 代数幾何学小研究集会
  • Place of Presentation: 埼玉大学
  • Year and Date: 2010-03-02
• [Presentation] Birational unboundedness of Q-Fano varieties of dimension at least five (2010)
  • Author(s): 岡田拓三
  • Organizer: 代数幾何小研究集会
  • Place of Presentation: 埼玉大学
  • Year and Date: 2010-03-02
• [Presentation] On the rationality problem of Q-Fano weighted hypersurfaces of dimension three (2010)
  • Author(s): 岡田拓三
  • Organizer: 代数幾何学セミナー
  • Place of Presentation: 北海道大学
  • Year and Date: 2010-01-25
• [Presentation] Birational unboundedness of Q-Fano varieties of dimension at least five (2009)
  • Author(s): 岡田拓三
  • Organizer: 高次元代数幾何の周辺
  • Place of Presentation: 数理解析研究所
  • Year and Date: 2009-12-18