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2006 Fiscal Year Final Research Report Summary

Research on Application of Computer Algebra to Algebraic Geometry

Research Project

Project/Area Number 15540024
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

MARUYAMA Masaki  Kyoto University, Graduate School of Science, Professor, 大学院理学研究科, 教授 (50025459)

Co-Investigator(Kenkyū-buntansha) MORIWAKA Atushi  Kyoto University, Graduate School of Science, Professor, 大学院理学研究科, 教授 (70191062)
KATO Fumiharu  Kyoto University, Graduate School of Science, Associate Professor, 大学院理学研究科, 助教授 (50294880)
Project Period (FY) 2003 – 2006
Keywordscomputer algebra / vector bundle / Tango bundle / stable vector bundle / projective space
Research Abstract

The existence and the construction problem of algebraic vector bundles has attracted many algebraic geometers, in connection with the classical existence problem of subvarieties. Stimulated by Weil's dream to genralarize the automorphic forms in terms of vector bundles, Grothendieck and Atiyah initiated the theory of algebraic vector bundles. Then Narasimhan, Seshadri, Mumford et al. have deeply studied the theory and have gone to the construction of the moduli spaces and their properties. Thanks to them, the foundation of the theory of algebraic vector bundles on curves has been settled though many serious problems are still remaining to be solved. Schwarzenberger began the study on algebraic vector bundles on algebraic surface and then the head investigator of this research project found a general way to construct algebraic vector bundles on higher dimensional varieties.
We have, however, no clear perspective about the existence and construction of low rank vector bundles on the projective spaces of dimension not less than four. In the present situation, it might be crucial to study the Tango bundle, which is essentially unique rank 2, indecomposable vector bundle on 5-dimensional projective space even though the ground field is of characteristic 2. In this project we set, therefore, our main target to study the Tango bundle by using Computer Algebra. We succeeded to represent the Tango bundle on Computer Algebra by a 15 x34 matrix whose entries are homogeneous quadratic forms in 6 variables. Watching this matrix we can determine the transition matrices of the Tango bundle and by using Computer Algebra we get a resolution of the Tango bundle by direct sums of line bundles. Then we can compute the Chern class of the Tango bundle. Shifting the first Chern class of the Tango bundle and computing (using Computer Algebra) the 0-th cohomology, we see that the Tango bundle is stable.

  • Research Products

    (9 results)

All 2006 2005 2004

All Journal Article (8 results) Book (1 results)

  • [Journal Article] On the finiteness of abelian varieties with bounded modular height2006

    • Author(s)
      森脇淳
    • Journal Title

      Adv. Std. in Pure Math. 45

      Pages: 157-187

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Arithmetic structure of CMSZ fake projective plane2006

    • Author(s)
      加藤 文元, 落合啓之
    • Journal Title

      J. of Algebra 305

      Pages: 1166-1185

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] On the finiteness of abelian varieties with bounded modular height2006

    • Author(s)
      A.Moriwaki
    • Journal Title

      Adv. Std. in Pure Math. 45

      Pages: 157-187

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Arithmetic structure of CMSZ fake protective plane2006

    • Author(s)
      F.Kato, H.Ochiai
    • Journal Title

      J. of Algebra 305

      Pages: 1166-1185

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Non-archimedean orbifolds covered by Mumford curves2005

    • Author(s)
      加藤文元
    • Journal Title

      J. of Alg. Geom 14

      Pages: 1-34

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Zur Entartung schwachverzweigter Gruppenoperationen auf Kurven2005

    • Author(s)
      G.Cornelissen, 加藤文元
    • Journal Title

      J. Reine Angew. Math. 589

      Pages: 201-236

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Non-archimedian orbifolds covered by Mumford curves2005

    • Author(s)
      F.Kato
    • Journal Title

      J. of Alg. Geom 14

      Pages: 1-34

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Zur Entartung schwachverzweigter Gruppenoperationen auf Kurven2005

    • Author(s)
      G.Cornelissen, F.Kato
    • Journal Title

      J. Reine Angew. Math. 589

      Pages: 201-236

    • Description
      「研究成果報告書概要(欧文)」より
  • [Book] 代数幾何学2004

    • Author(s)
      廣中平祐 講義, 森重文 記録, 丸山正樹, 森脇淳, 川口周 改訂・加筆・編集
    • Total Pages
      175
    • Publisher
      京都大学学術出版会
    • Description
      「研究成果報告書概要(和文)」より

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Published: 2008-05-27  

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