Study on toric Fano varieties and Calabi-Yau hypersurfaces

• Project/Area Number: 26400034
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Tohoku University
• Principal Investigator: OGATA Shoetsu 東北大学, 理学研究科, 准教授 (90177113)
• Research Collaborator: ZHAO Huai-Liang
• Project Period (FY): 2014-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2016: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2014: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
• Keywords: 代数幾何学 / トーリック多様体 / カラビ・ヤウ多様体 / ファノ多様体

Outline of Final Research Achievements

This study is aimed to find projective geometric properties of toric Fano varieties and their Calabi-Yau hypersurfaces by using the corresponding lattice polytopes. We obtain a characterization of toric Fano varieties with higher index among projective toric varieties. And we find out a projective property of Calabi-Yau hypersurfaces in toric Fano 4-folds by showing the normality of lattice polytopes corresponding toric weak Fano 3-folds. We publish these results as research papers and give talks at workshops held in Japan and overseas.

Research Products

• [Journal Article] A characterization of Gorenstein toric Del Pezzo n-folds (2015)
  • Author(s): 尾形庄悦 趙懐亮
  • Journal Title: Far East Journal of Mathematical Sciences Volume: 96
  • Peer Reviewed
• [Journal Article] A Characterization of Gorenstein Toric Del Pezzo n-folds (2015)
  • Author(s): 尾形庄悦, 超懐亮
  • Journal Title: Far East Journal of Mathematical Sciences Volume: 96 Pages: 1049-1064
  • Peer Reviewed
• [Journal Article] Nef and Big Divisors on Toric Weak Fano 3-Folds (2014)
  • Author(s): 尾形庄悦
  • Journal Title: International Mathematical Forum Volume: 9 Pages: 661-675
  • Peer Reviewed / Open Access
• [Journal Article] A Characterization of Gorenstein Toric Fano n-folds with Index n and Fujita's Conjecture (2014)
  • Author(s): 尾形庄悦, 超懐亮
  • Journal Title: Far East Journal of Mathematical Sciences Volume: 94 Pages: 65-88
  • Peer Reviewed
• [Presentation] On ideals defining toric 3-folds (2017)
  • Author(s): 尾形 庄悦
  • Organizer: 農工大数学セミナー2017
  • Place of Presentation: 東京農工大学
  • Year and Date: 2017-03-21
• [Presentation] On ideals defining toric varieties (2016)
  • Author(s): 尾形 庄悦
  • Organizer: EMS Summer School on Multigraded Algebra and Applications
  • Place of Presentation: Moieciu de Sus, Romania
  • Year and Date: 2016-08-17
• [Presentation] On ideals defining toric varieties (2015)
  • Author(s): 尾形庄悦
  • Organizer: KAIST 代数幾何セミナー
  • Place of Presentation: KAIST(韓国科学技術院)、大田(韓国)
  • Year and Date: 2015-09-22
• [Presentation] Nef and bid line bundles on weak Fano toric 3-folds (2014)
  • Author(s): 尾形庄悦
  • Organizer: 研究集会「代数多様体とその周辺」
  • Place of Presentation: 琉球大学（沖縄県中頭郡西原町）
  • Year and Date: 2014-09-29 – 2014-10-02
• [Presentation] Projective normality of Toric weak Fano 3-folds (2014)
  • Author(s): 尾形庄悦
  • Organizer: Vector Bundles on Algebraic Curves--Algebraic Varieties: Bundles, Topology, Physics
  • Place of Presentation: ベルリン自由大学（ドイツ）
  • Year and Date: 2014-09-01 – 2014-09-05
• [Presentation] Projective normality of toric weak Fano 3-folds (2014)
  • Author(s): 尾形庄悦
  • Organizer: ミニワークショップ Calabi-Yau varities: Arithmetic, Geometry and Physics
  • Place of Presentation: 津田塾大学（東京都小平市）
  • Year and Date: 2014-08-07 – 2014-08-08