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Explicit Study of Algebraic Varieties

Research Project

Project/Area Number 18540001
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionHokkaido University

Principal Investigator

SHIMADA Ichiro  Hokkaido University, Fac. of .Sci., Associate Professor (10235616)

Co-Investigator(Kenkyū-buntansha) OKA Mutsuo  Tokyo University of Science, Grad. school of sci, Professor (40011697)
ISHIKAAWA Goo  Hokkaido University, Fac. of.Sci, Professor (50176161)
KONNO Kazuhiro  Osaka University, Grad school of sci., Professor (10186869)
SAITO Mutsuni  Hokkaido University, Fac. of.Sci,Asso, Professor (70215565)
TOKUNAGA Hiroo  Tokyo Metropolitan University, Grad. school of science and engineering, Professor (30211395)
Project Period (FY) 2006 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥3,920,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥420,000)
Fiscal Year 2007: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Fiscal Year 2006: ¥2,100,000 (Direct Cost: ¥2,100,000)
KeywordsK3 surface / plane curve / lattices / supersingularity / rational double point / fundamental group / dual variety / K3曲面 / 平面曲線 / 分岐被覆 / 基本群 / グレブナー基底 / 格子 / 6次曲線 / 超特異性 / 正標数
Research Abstract

(1) By a joint work with Due Tai Pho, we proved the unirationality of supersingular K3 surfaces in characteristic 5 with Artin invariant 〓 3.
(2) We classified all possible configurations of rational double points on complex normal algebraic K3 surfaces, and discussed the same classification for supersingular K3 surfaces in sufficiently large characteristics.
(3) We investigated the set of isomorphism classes of transcendental lattices of complex algebraic varieties obained from a single algebraic variety defined over a number field by various embeddings of the base field into the complex number field, and produced many examples of non-homeomorphic complex algebraic varieties that are conjugate under the automorphism of the complex number field.
In particular, we investigate the case of singular. K3 surfaces by means of the class field theory of imaginary quadratic fields. As an application, we obtained a lower bound of the degree of the base field of singular K3 surfaces.
(4) Suppose that we obtain a supersingular K3 surface X (P) by reduction at a finite place P of a base field of a singular K3 surface X defined over a number field. We investigated the orthogonal complement of the Neron-Severi lattice of X in that of X (P), and proved an analogous result as the case of the transvendental lattices.
(5) We proved a Lefschetz hyperplane section theorem for the topological fundamental group of the complement of the duial variety in the Grassmannian variety, and invetigate the Zariski-van Kampen type relation of that fundamental group with the barid group of a punctured Riemann surface.

Report

(3 results)
  • 2007 Annual Research Report   Final Research Report Summary
  • 2006 Annual Research Report
  • Research Products

    (54 results)

All 2008 2007 2006

All Journal Article (21 results) (of which Peer Reviewed: 12 results) Presentation (31 results) Book (2 results)

  • [Journal Article] On arithmetic Zariski pairs in degree 62008

    • Author(s)
      I.Shimada
    • Journal Title

      Adv. Geom. (to appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Transcendental lattices and supersingular reduction lattices of a singular K3 surface2008

    • Author(s)
      I.Shimada
    • Journal Title

      Trans. Amer. Math. Soc. (to appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] On arithmetic Zariski pairs in degree 62008

    • Author(s)
      I. Shimada
    • Journal Title

      Adv. Geom. (to appear)

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Transcendental lattices and Supersingular reduction lattices of asingular K3 surface2008

    • Author(s)
      I. Shimada
    • Journal Title

      Trans. Amer. Math. Soc. (to appear)

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On Kummer type construction of supersingular K3 surfaces in characteristic 2.2007

    • Author(s)
      I.Shimada, De-Qi Zhang
    • Journal Title

      Pacific J. Math. 232

      Pages: 379-400

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] On normal K3 surfaces2007

    • Author(s)
      I.Shimada
    • Journal Title

      Michigan Math. J. 55

      Pages: 395-416

    • NAID

      120000949903

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] K3 surfaces with ten cusps.2007

    • Author(s)
      I.Shimada, De-Qi Zhang
    • Journal Title

      Contemp. Math. 422

      Pages: 187-211

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] On Kummer type construction of supersingular K3 surfaces in characteristic 22007

    • Author(s)
      I. Shimada, De-Qi Zhang
    • Journal Title

      Pacific J. Math. 232

      Pages: 379-400

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] On normal K3 surfaces2007

    • Author(s)
      I. Shimada
    • Journal Title

      Michigan Math. J. 55

      Pages: 395-416

    • NAID

      120000949903

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Journal Article] K3 surfaces with ten cusps2007

    • Author(s)
      I. Shimada, De-Qi Zhang
    • Journal Title

      Contemp. Math. 422

      Pages: 187-211

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] On Kummer type construction of supersingular K3 surfaces in characteristic 2.2007

    • Author(s)
      I. Shimada, De-Qi Zhang
    • Journal Title

      Pacific J. Math. 232

      Pages: 379-400

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] K3 surfaces with ten cusps.2007

    • Author(s)
      I. Shimada, De-Qi Zhang
    • Journal Title

      Contemp. Math. 422

      Pages: 187-211

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Unirationality of certain supersingular $K3$ surfaces in characteristic 5.2006

    • Author(s)
      I. Shimada, Duc Tai Pho
    • Journal Title

      Manuscripta Math. 121

      Pages: 425-435

    • NAID

      120000963716

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Singularities of dual varieties in characteristic 32006

    • Author(s)
      I. Shimada
    • Journal Title

      Geom. Dedicata 120

      Pages: 141-177

    • NAID

      120000964075

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Moduli curves of supersingular $K3$ surfaces in characteristic 2 with Artin invariant 22006

    • Author(s)
      I. Shimada
    • Journal Title

      Proc. Edinb. Math. Soc. 49

      Pages: 435-503

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Unirationality of certain supersingular K3 surfaces in char-acteristic 52006

    • Author(s)
      I. Shimada, Duc Tai Pho
    • Journal Title

      Manuscripta Math. 121

      Pages: 425-43

    • NAID

      120000963716

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Singularities of dual varieties in characteristic 32006

    • Author(s)
      I. Shimada
    • Journal Title

      Geom. Dedicata. 120

      Pages: 141-177

    • NAID

      120000964075

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Moduli curves of supersingular $K3$ surfaces in characteristic 2 with Artin invariant 22006

    • Author(s)
      I. Shimada
    • Journal Title

      Proc. Edinb. Math. Soc. (2)49

      Pages: 435-503

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Singularities of dual varieties in characteristic 32006

    • Author(s)
      Ichiro Shimada
    • Journal Title

      Geom. Dedicata 120

      Pages: 141-177

    • NAID

      120000964075

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Moduli curves of supersingular K3 surfaces in characteristic 2 with Artin invariant 22006

    • Author(s)
      Ichiro Shimada
    • Journal Title

      Proc. Edinb. Math. Soc 49

      Pages: 435-503

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Unirationality of certain supersingular K3 surfaces in characteristic 52006

    • Author(s)
      D.T.Pho
    • Journal Title

      manuscripta math. 121

      Pages: 425-435

    • Related Report
      2006 Annual Research Report
  • [Presentation] Dessins d'enfants and transcendental lattices of singular K3 surfaces2008

    • Author(s)
      I. Shimada
    • Organizer
      酒井文雄先生還暦記念研究集会
    • Place of Presentation
      埼玉大学理学部数学教室
    • Year and Date
      2008-03-10
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] Dessins d'enfants and transcendental lattices of singular K3 surfaces2008

    • Author(s)
      Ichiro Shimada
    • Organizer
      Symposium for Professor Fumio Sakai's 60th birthday
    • Place of Presentation
      Saitama University
    • Year and Date
      2008-03-10
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Topology and arithmetic of maximizing sextics2008

    • Author(s)
      I. Shimada
    • Organizer
      SNU-HU 5th Joint Symposium on Mathematics
    • Place of Presentation
      Seoul National University
    • Year and Date
      2008-01-25
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] Singular K3 surfaces and non-homeomorphic conjugate varieties2007

    • Author(s)
      Ichiro Shimada
    • Organizer
      Algebraic Geometry and Com-mutative Algebra Tokyo 2007
    • Place of Presentation
      University of Tokyo
    • Year and Date
      2007-12-14
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Singular K3 surfaces and non-homeomorphic conjugate varieties2007

    • Author(s)
      I. Shimada
    • Organizer
      Algebraic Geometry and Commutative Algebra Tokyo 2007
    • Place of Presentation
      東京大学大学院数理科学研究科
    • Year and Date
      2007-11-14
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] Fundamental groups of complements of dual varieties in Grassmannian2007

    • Author(s)
      I. Shimada
    • Organizer
      高次元代数多様体とベクトル束の代数幾何学
    • Place of Presentation
      九州大学理学部,博多
    • Year and Date
      2007-09-11
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] Fundamental groups of complements of dual varieties in Grassmannian2007

    • Author(s)
      Ichiro Shimada
    • Organizer
      Algebraic Geometry of Higher Dimensional Varieties and Vec-tor Bundles
    • Place of Presentation
      Kyushu University
    • Year and Date
      2007-09-11
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Tanscendental lattices and supersingular reduction lattces of a singular K3 surface2007

    • Author(s)
      I. Shimada
    • Organizer
      COE seminar on algebraic geometly
    • Place of Presentation
      慶応大学日吉キャンパス
    • Year and Date
      2007-09-04
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Transcendental lattices and supersingular reduction lattices of a singular K3 surface2007

    • Author(s)
      Ichiro Shimada
    • Organizer
      COE seminar on algebraic geom-etry
    • Place of Presentation
      Keio University
    • Year and Date
      2007-09-04
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Transcendental lattices and supersingular reduction lattices of asingular K3 surface2007

    • Author(s)
      I. Shimada
    • Organizer
      COE seminar on algebraic geometry
    • Place of Presentation
      慶応大学日吉キャンパス
    • Year and Date
      2007-09-04
    • Related Report
      2007 Annual Research Report
  • [Presentation] Non-homeomorphic conjugate complex varieties2007

    • Author(s)
      I. Shimada
    • Organizer
      Singularities in Geometry and Topology
    • Place of Presentation
      立山国際ホテル
    • Year and Date
      2007-08-30
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] Non-homeomorphic conjugate complex varieties2007

    • Author(s)
      Ichiro Shimada
    • Organizer
      Singularities in Geometry and Topology
    • Place of Presentation
      Tateyama Hotel
    • Year and Date
      2007-08-30
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Rational double points on K3 surfaces2007

    • Author(s)
      I. Shimada
    • Organizer
      Niigata Workshop on Complex Geometry and Singularities
    • Place of Presentation
      CrossPal Niigata
    • Year and Date
      2007-08-24
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] Rational double points on K3 surfaces2007

    • Author(s)
      Ichiro Shimada
    • Organizer
      Niigata Workshop on Complex Geometry and Singularities
    • Place of Presentation
      Cross Pal Niigata
    • Year and Date
      2007-08-24
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 数体上定義された特異K3曲面2007

    • Author(s)
      I. Shimada
    • Organizer
      第52回代数学シンポジウム
    • Place of Presentation
      神戸大学
    • Year and Date
      2007-08-08
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Singular K3 surfaces over a number field2007

    • Author(s)
      Ichiro Shimada
    • Organizer
      The 52-nd Symposium for alge-bra
    • Place of Presentation
      Kobe University
    • Year and Date
      2007-08-08
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 数体上定義された特異 K3 曲面2007

    • Author(s)
      I. Shimada
    • Organizer
      第52回代数学シンポジウム
    • Place of Presentation
      神戸大学
    • Year and Date
      2007-08-08
    • Related Report
      2007 Annual Research Report
  • [Presentation] Fundamental groups of complements of dual varieties in Grassmannian2007

    • Author(s)
      I. Shimada
    • Organizer
      Conference on Symplectic Geometry
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2007-07-26
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] Fundamental groups of complements of dual varieties in Grassmannian2007

    • Author(s)
      Ichiro Shimada
    • Organizer
      Conference on Symplectic Geom-etry
    • Place of Presentation
      RIMS
    • Year and Date
      2007-07-26
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Non-homeomorphic conjugate complex varieties2007

    • Author(s)
      I. Shimada
    • Organizer
      6th Polish-Japanese Singularity Theory Working Days
    • Place of Presentation
      Sopot,Poland
    • Year and Date
      2007-07-12
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Non-homeomorphic conjugate complex varieties2007

    • Author(s)
      Ichiro Shimada
    • Organizer
      The 6th Polish-Japanese Singu-larity Theory Working Days
    • Place of Presentation
      Sopot, Poland
    • Year and Date
      2007-07-12
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Non-homeomorphic conjugate complex varieties2007

    • Author(s)
      I. Shimada
    • Organizer
      6th Polish-Japanese Singularity Theory Working Days
    • Place of Presentation
      Sopot, Poland
    • Year and Date
      2007-07-12
    • Related Report
      2007 Annual Research Report
  • [Presentation] 分岐曲線の補集合の基本群とリーマン面の単純ブレイド群2007

    • Author(s)
      I. Shimada
    • Organizer
      第1回「分岐被覆に関連する代数幾何とトポロジー」ワークショップ
    • Place of Presentation
      東北学院大学工学部多賀城キャンパス
    • Year and Date
      2007-06-09
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Simple braid group of Riemann surface and the fundamental group of the complement of the branch curves2007

    • Author(s)
      Ichiro Shimada
    • Organizer
      Workshop for ramified coverings
    • Place of Presentation
      Tohoku Gakuin University
    • Year and Date
      2007-06-09
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 分岐曲線の補集合の基本群とリーマン面の単純ブレイド群2007

    • Author(s)
      I. Shimada
    • Organizer
      第1回「分岐被覆に関連する代数幾何とトポロジー」ワークショップ
    • Place of Presentation
      東北学院大学工学部多賀城キャンバス
    • Year and Date
      2007-06-09
    • Related Report
      2007 Annual Research Report
  • [Presentation] On arithmetic Zariski pairs in degree 62006

    • Author(s)
      Ichiro Shimada
    • Organizer
      特異点と基本群のワークショップ
    • Place of Presentation
      東京理科大学
    • Year and Date
      2006-12-27
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On arithmetic Zariski pairs in degree 62006

    • Author(s)
      Ichiro Shimada
    • Organizer
      Workshop for singularities and fundamental groups
    • Place of Presentation
      Tokyo University of Science
    • Year and Date
      2006-12-27
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On arithmetic Zariski pairs in degree 62006

    • Author(s)
      Ichiro Shimada
    • Organizer
      Mini-workshop on hyperplane arrangements
    • Place of Presentation
      北海道大学
    • Year and Date
      2006-12-07
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On arithmetic Zariski pairs in degree 62006

    • Author(s)
      Ichiro Shimada
    • Organizer
      Mini-workshop on hyperplane arrangements
    • Place of Presentation
      Hokkaido University
    • Year and Date
      2006-12-07
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Singularities of dual varieties in characteristic 32006

    • Author(s)
      Ichiro Shimada
    • Organizer
      International Symposium on singularity theory and its applications
    • Place of Presentation
      Changsha, 中国
    • Year and Date
      2006-09-07
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Singularities of dual varieties in characteristic 32006

    • Author(s)
      Ichiro Shimada
    • Organizer
      International Symposium on singularity theory and its applications
    • Place of Presentation
      Changsha, China
    • Year and Date
      2006-09-07
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Book] On arithmetic Zariski pairs in degree 62008

    • Author(s)
      I. Shimada
    • Publisher
      Adv. Geom.
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Book] Transcendental lattices and supersingular reduction lattices of a singular K3 surface2008

    • Author(s)
      I. Shimada
    • Publisher
      Trans. Amer. Math. Soc.
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary

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Published: 2006-04-01   Modified: 2016-04-21  

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