Algebraic geometry of hypersurfaces and lines in projective spaces and Hodge structure

• Project/Area Number: 20740014
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Algebra
• Research Institution: Tokyo Denki University (2011); Osaka University (2008-2010)
• Principal Investigator: IKEDA Atushi (IKEDA Atsushi) 東京電機大学, 工学部, 准教授 (40397617)
• Project Period (FY): 2008 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000); Fiscal Year 2011: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2010: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2009: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2008: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: 代数幾何 / ホッジ構造 / 3次曲面 / 周期写像 / 3次超曲面

Research Abstract

We described geometric property of a hypersurface in a projective space by using the Hodge structure of the variety of lines which intersect the hypersurface with some multiplicity. We explained the relation between geometric property of lines in cubic surface and the Neron-Severi lattice of the variety of lines, and we computed the structure of the Neron-Severi lattice. And we showed the injectivity of the period map which is defined by the Hodge structure of variety of lines form the family of cubic surfaces over the moduli space.

Research Products

• [Journal Article] A period map for cubic surfaces (2011)
  • Author(s): 池田京司
  • Journal Title: 数理解析研究所講究録 Volume: 1745 Pages: 5-10
• [Journal Article] Subvarieties of generic hypersurfaces in a nonsingular projective toric variety (2009)
  • Author(s): 池田京司
  • Journal Title: Mathematische Zeitschrift 263 Pages: 923-937
  • Peer Reviewed
• [Journal Article] The varieties of intersect-ions of lines and hypersurfaces in projective spaces, "Higher dimensi-onal algebraic varieties and vector bundles" (2008)
  • Author(s): 池田京司
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: B9 Pages: 115-125
  • Peer Reviewed
• [Journal Article] The varieties of intersections of lines and hypersurfaces in projective spaces (2008)
  • Author(s): 池田京司
  • Journal Title: Higher dimensional algebraic varieties and vector bundles RIMS Kokyuroku Bessatsu B9 Pages: 115-125
  • Peer Reviewed
• [Journal Article] The varieties of intersections of lines and hypersurfaces in projective spaces (2008)
  • Author(s): 池田京司
  • Journal Title: RIMS Kokyuroku Bessatsu, Higher dimensional algebraic varieties and vector bundles B9 Pages: 115-125
  • Peer Reviewed
• [Journal Article] The double cover of cubic surfaces branched along their Hessian
  • Author(s): 池田京司
  • Journal Title: arXiv mathematics
  • URL: http://arxiv.org/PS_cache/arxiv/pdf/1012/1012.4242v1.pdf
• [Journal Article] The varieties of tangent lines to hypersurfacese in projective spaces
  • Author(s): 池田京司
  • Journal Title: arXiv mathematics
  • URL: http://arxiv.org/PS_cache/arxiv/pdf/1012/1012.2186v1.pdf
• [Journal Article] The double cover of cubic surfaces branched along their Hessian
  • Author(s): 池田京司
  • URL: http://arxiv.org/PS_cache/arxiv/pdf/1012/1012.4242vl.pdf
• [Journal Article] The varieties of tangent lines to hypersurfacese in projective spaces
  • Author(s): 池田京司
  • URL: http://arxiv.org/PS_cache/arxiv/pdf/1012/1012.2186vl.pdf
• [Presentation] 3次超曲面の幾何とホッジ構造 (2011)
  • Author(s): 池田京司
  • Organizer: 代数学シンポジウム
  • Place of Presentation: 岡山大学環境理工学部
  • Year and Date: 2011-08-10
• [Presentation] 3次超曲面の幾何とホッジ構造 (2011)
  • Author(s): 池田京司
  • Organizer: 代数学シンポジウム
  • Place of Presentation: 岡山大学(岡山市)
  • Year and Date: 2011-08-10
• [Presentation] 次超曲面のEckardt点について (2010)
  • Author(s): 池田京司
  • Organizer: 射影多様体の幾何とその周辺2010
  • Place of Presentation: 高知大学理学部
  • Year and Date: 2010-11-06
• [Presentation] 3次超曲面のEckardt点について (2010)
  • Author(s): 池田京司
  • Organizer: 射影多様体の幾何とその周辺2010
  • Place of Presentation: 高知大学理学部
  • Year and Date: 2010-11-06
• [Presentation] Nodal cubic surfaces and Eisenstein series (2010)
  • Author(s): 池田京司
  • Organizer: 特異点と多様体の幾何学
  • Place of Presentation: 山形大学理学部
  • Year and Date: 2010-09-18
• [Presentation] Nodal cubic surfaces and Eisenstein series (2010)
  • Author(s): 池田京司
  • Organizer: 特異点と多様体の幾 何学
  • Place of Presentation: 山形大学理学部
  • Year and Date: 2010-09-18
• [Presentation] Nodal cubic surfaces and Eisenstein series (2010)
  • Author(s): 池田京司
  • Organizer: 代数幾何の関連する諸分野
  • Place of Presentation: 北海道大学理学部
  • Year and Date: 2010-08-30
• [Presentation] Nodal cubic surfaces and Eisenstein series (2010)
  • Author(s): 池田京司
  • Organizer: 代数幾何の関連する 諸分野
  • Place of Presentation: 北海道大学理学部
  • Year and Date: 2010-08-30
• [Presentation] Nodal cubic surfaces and generalized Kummer surfaces (2010)
  • Author(s): 池田京司
  • Organizer: 複素幾何と代数幾何の若手研究集会
  • Place of Presentation: 熊本大学理学部
  • Year and Date: 2010-02-19
• [Presentation] Nodal cubic surfaces and generalized Kummer surfaces (2010)
  • Author(s): 池田京司
  • Organizer: 複素幾 何と代数幾何の若手研究集会
  • Place of Presentation: 熊本大学理学部
  • Year and Date: 2010-02-19
• [Presentation] A period map for cubic surfaces (2009)
  • Author(s): 池田京司
  • Organizer: Hodge理論と代数幾何学
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2009-06-29
• [Presentation] Aperiod map fbr cubic surfaces, Hodge (2009)
  • Author(s): 池田京司
  • Organizer: 理論と代数幾何学
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2009-06-29
• [Presentation] Periods and geometry of cubic forms (2008)
  • Author(s): 池田京司
  • Organizer: Geometry of singularities and manifolds
  • Place of Presentation: 草津セミナーハウス
  • Year and Date: 2008-09-13