K3 surfaces and related algebraic varieties

2010 Fiscal Year Self-evaluation Report

• Project/Area Number: 20340002
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Hiroshima University
• Principal Investigator: SHIMADA Ichiro Hiroshima University, 大学院・理学研究科, 教授 (10235616)
• Project Period (FY): 2008 – 2011
• Keywords: K3曲面 / 6次曲線 / 基本群 / 格子理論 / 超越格子 / 超特異性

Research Abstract

本研究の目的は,K3曲面に関する多種多様な問題に対して次の研究活動を行うことにより,従来の研究をさらに大規模に発展させるとともに異なる研究方法の間との相互作用をうながすことである
まず,格子理論的計算に還元できる部分を確定する.還元できる部分に対しては,汎用性の高いアルゴリズムを書いて計算機に実行させる,できるだけ多数の研究集会に参加して格子理論と計算機の組み合わせの強力さを宣伝し,多くの研究者を格子理論と計算機によるK3曲面の研究に巻き込むことを目指す
還元できない部分,とくに具体的な定義方程式を必要とする問題に対しては,グレブナー基底を用いたアプローチを試みる.さらに,格子理論的データ以外の量(クリスタル周期,ブラウア群など)を計算機で取り扱う手法を開発する

Research Products

• [Journal Article] Primitivity of sublattices generated by classes of curves on an algebraic surface. (2010)
  • Author(s): I. Shimada, N. Takahashi
  • Journal Title: Comment.Math.Univ.St.Pauli 59 Pages: 77-95
  • Peer Reviewed
• [Journal Article] Lattice Zariski k-pies of plane sextic curves and Z-splitting curves for double plane sextics. (2010)
  • Author(s): I. Shimada
  • Journal Title: Michigan Math.J. 59, no.3 Pages: 621-665
  • Peer Reviewed
• [Journal Article] Topology of curves on a surface and lattice-theoretic invariants of coverings of the surface. (2010)
  • Author(s): I. Shimada
  • Journal Title: Algebraic geometry in East Asia-Seoul 2008, Adv.Stud.Pure Math., Math.Soc.Japan 60 Pages: 361-382
  • Peer Reviewed
• [Journal Article] Generalized Zariski-van Kampen theorem and its application to Grassmannian dual varieties. (2010)
  • Author(s): I. Shimada
  • Journal Title: Internat.J.Math. 21, no.5 Pages: 591-637
  • Peer Reviewed
• [Journal Article] Non-homeomorphic conjugate complex varieties. (2009)
  • Author(s): I. Shimada
  • Journal Title: Singularities Niigata-Toyama 2007, Adv.Stud.Pure Math., Math.Soc.Japan 56 Pages: 285-301
  • Peer Reviewed
• [Presentation] Lattices of algebraic cycles in positive characteristics (2010)
  • Author(s): 島田伊知朗
  • Organizer: 5th Pacific Rim Conference on Mathematics
  • Place of Presentation: Stanford University, USA
  • Year and Date: 2010-07-02
• [Presentation] Supersingular K3 surfaces and lattice theory (2010)
  • Author(s): I. Shimada
  • Organizer: Arithmetic and Algebraic Geometry 2011
  • Place of Presentation: 東京大学数理科学研究科
  • Year and Date: 2010-01-21
• [Presentation] On lattice-invariants of complex algebraic surfaces and their applications (2009)
  • Author(s): I. Shimada
  • Organizer: Symposium in algebraic geometry
  • Place of Presentation: Busan, Korea
  • Year and Date: 2009-12-28
• [Presentation] On lattice-theoretic invariants of curves on a surface (2009)
  • Author(s): I. Shimada
  • Organizer: The 2nd MSJ-ST ArranaPments of Hypernlanes
  • Place of Presentation: 北海道大学
  • Year and Date: 2009-08-12
• [Presentation] Zariski pairs and lattice theory (2009)
  • Author(s): 島田伊知朗
  • Organizer: Antalya Algebra Days XI
  • Place of Presentation: Antalya, Turk
  • Year and Date: 2009-03-24
• [Remarks] ホームページ
  • URL: http://www.math.sci.hiroshima-u.ac.jp/~shimada/index.html