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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)
並河 良典  京都大学, 大学院・理学研究科, 助教授 (80228080)
石井 亮  京都大学, 大学院・工学研究科, 講師 (10252420)
Project Period (FY) 2003 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2006: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2005: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2004: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2003: ¥1,100,000 (Direct Cost: ¥1,100,000)
Keywordscomputer algebra / vector bundle / Tango bundle / stable vector bundle / projective space / 変形 / コホモロジー / 還元列 / 障害理論 / Singular / 安定層 / 丹後バンドル / ドナルドソン多項式 / モデュライ
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.

Report

(5 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • Research Products

    (20 results)

All 2006 2005 2004 Other

All Journal Article (16 results) Book (2 results) Publications (2 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
    • Related Report
      2006 Final Research Report Summary
  • [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
    • Related Report
      2006 Final Research Report Summary
  • [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
    • Related Report
      2006 Final Research Report Summary
  • [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
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] On the finiteness of abelian varieties with bounded modular height2006

    • Author(s)
      森脇 淳
    • Journal Title

      Advanced Studies in Pure Mathematics 45

      Pages: 157-187

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Arithmetic structure of CMSZ fake projective plane2006

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

      Journal of Algebra 305

      Pages: 1166-1185

    • Related Report
      2006 Annual Research Report
  • [Journal Article] On the finiteness of abelian varieties with bounded modular height2006

    • Author(s)
      森脇 淳
    • Journal Title

      Advanced Studies in Pure Mathematics (掲載予定)

    • Related Report
      2005 Annual Research Report
  • [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
    • Related Report
      2006 Final Research Report Summary
  • [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

    • NAID

      120006581747

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [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
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Zur Entartung schwachverzweigter Gruppenoperationen auf Kurven2005

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

      J. Reine Angew. Math. 589

      Pages: 201-236

    • NAID

      120006581747

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Non-archimedean orbifolds covered by Mumford curves2005

    • Author(s)
      加藤文元
    • Journal Title

      Journal of Alg.Geom. 14

      Pages: 1-34

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Zur Entartung schwachverzweigter Gruppenoperationen auf Kurven2005

    • Author(s)
      Gunther Cornelissen, 加藤 文元
    • Journal Title

      J.Reine Angew.Math. 589

      Pages: 201-236

    • NAID

      120006581747

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Diophantine geometry Vvewed from Arakelov geometry2004

    • Author(s)
      森脇 淳
    • Journal Title

      Sugaku Expositions 17 2

      Pages: 219-233

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Flops of G-Hilb and equivalences of derived categories by variation of GIT quotient2004

    • Author(s)
      石井 亮
    • Journal Title

      Duke Math.J. 124

      Pages: 259-307

    • Related Report
      2004 Annual Research Report
  • [Journal Article] The number of algebraic cycles with bounded degree

    • Author(s)
      森脇 淳
    • Journal Title

      J.of Mathematics of Kyoto Univ. (掲載予定)

    • Related Report
      2004 Annual Research Report
  • [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
    • Related Report
      2006 Final Research Report Summary
  • [Book] 代数幾何学2004

    • Author(s)
      廣中平祐 著, 丸山正樹他編
    • Total Pages
      182
    • Publisher
      京都大学学術出版会
    • Related Report
      2004 Annual Research Report
  • [Publications] 山田紀美子: "A sequence of blowing-ups connecting moduli of sheaves and The Donalson polynomial under change of polarization"Journal of Mathematics of Kyoto University. 43・4. (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] 並河良典: "Mukai flops and derived categories"Journal fur die reine und angewandte Mathematik. 560. 65-76 (2003)

    • Related Report
      2003 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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