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Algebraic group actions and the structure of affine algebraic varieties

Research Project

Project/Area Number 15540043
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionUniversity of Hyogo (2004-2005)
Himeji Institute of Technology (2003)

Principal Investigator

MASUDA Kayo  University of Hyogo, Graduate school of Material Science, Associated Professor, 大学院・物質理学研究科, 助教授 (40280416)

Co-Investigator(Kenkyū-buntansha) MIYANISHI Masayoshi  Kwansei Gakuin University, School of Science and Technology, Professor, 理工学部, 教授 (80025311)
Project Period (FY) 2003 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥2,700,000 (Direct Cost: ¥2,700,000)
Fiscal Year 2005: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2004: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2003: ¥1,100,000 (Direct Cost: ¥1,100,000)
Keywordsaffine pseudo-plane / Cancellation Problem / Makar-Limanov invariant / Linearization Problem / Jacobian Conjecture / A^1-fibration / algebraic torus action / Cancellation Problem / Makar-Limanov invariant / A^1-fibration / 埋め込み問題 / G_m作用 / G endonlorphism / Qホモロジー平面
Research Abstract

1.Linearization Problem
We have had some results on the Cancellation Problem which is closely related to the Linearization Problem.
(1)There exists an infinite dimensional family of affine pseudo-planes without cancellation property. Furthermore, tom Dieck surfaces are characterized as affine pseudo-planes with nontrivial actions of an algebraic torus.
(2)The kernel of a triangular derivation with a slice is a polynomial ring.
2.Generalized Jacobian Problem
(1)We studied the Jacobian Problem for smooth affine surfaces with A^1-fibrations, especially, affine pseudo-planes. We succeeded in making clear of the structure of affine pseudo-planes to some extent with joint work with R.V.Gurjar and P.Russell, but could not resolve the Generalized Jacobian Problem for affine pseudo-planes.
(2)Miyanishi showed that the Generalized Jacobian Problem for affine pseudo-planes has an affirmative answer if they satisfy the generalized Sard property.

Report

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

    (46 results)

All 2005 2004 2003 Other

All Journal Article (40 results) Book (2 results) Publications (4 results)

  • [Journal Article] Equivariant cancellation for algebraic varieties2005

    • Author(s)
      K.Masuda, M.Miyanishi
    • Journal Title

      Proceedings of the RSME/AMS meeting in Comtem. Math. 369

      Pages: 183-195

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Affine pseudo-planes and cancellation problem2005

    • Author(s)
      K.Masuda, M.Miyanishi
    • Journal Title

      Transactions of the Amer. Math. Soc. 357

      Pages: 4867-4883

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Automorphisms of affine surfaces with A^1-fibrations2005

    • Author(s)
      R.V.Gurjar, M.Miyanishi
    • Journal Title

      Michigan Math. J. 53

      Pages: 33-55

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On two recent views of the Jacobian Conjecture2005

    • Author(s)
      T.Kambayashi, M.Miyanishi
    • Journal Title

      Proceedings of the RSME/AMS meeting in Comtem. Math. 369

      Pages: 113-138

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Affine pseudo-converings of algebraic surfaces2005

    • Author(s)
      M.Miyanishi
    • Journal Title

      J. Algebra 194

      Pages: 156-176

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Equivariant cancellation for algebraic varieties2005

    • Author(s)
      K.Masuda, M.Miyanishi
    • Journal Title

      Proceedings of the RSME/AMS meeting in Comtem.Math. 369

      Pages: 183-195

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Affine pseudo-planes and cancellation problem2005

    • Author(s)
      K.Masuda, M.Miyanishi
    • Journal Title

      Transactions of the Amer.Math.Soc. 357

      Pages: 4867-4883

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Automorphisms of affine surfaces with A^1-fibrations2005

    • Author(s)
      R.V.Gurjar, M.Miyanishi
    • Journal Title

      Michigan Math.J. 53

      Pages: 33-55

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On two recent views of the Jacobian Conjecture2005

    • Author(s)
      T.Kambayashi, M.Miyanishi
    • Journal Title

      Proceedings of the RSME/AMS meeting in Comtem.Math. 369

      Pages: 113-138

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Affine pseudo-coverings of algebraic surfaces2005

    • Author(s)
      M.Miyanishi
    • Journal Title

      J.Algebra 194

      Pages: 156-176

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Equivariant cancellation for algebraic varieties2005

    • Author(s)
      K.Masuda, M.Miyanishi
    • Journal Title

      Proceedings in Contemporary Math 369

      Pages: 183-195

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Automorphisms of affine surfaces with A^1-flbrat ions2005

    • Author(s)
      R.V.Gurjar, M.Miyanishi
    • Journal Title

      Michigan Math. J 53

      Pages: 33-55

    • Related Report
      2005 Annual Research Report
  • [Journal Article] On two recent views of the Jacob1an Conjecture2005

    • Author(s)
      T.Kambayashi, M.Miyanishi
    • Journal Title

      Proceedings in Contemporary Math 369

      Pages: 113-138

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Affine pseudo-coverings of algebraic surfaces2005

    • Author(s)
      M.Miyanishi
    • Journal Title

      J. Algebra 194

      Pages: 156-176

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Affine pseudo-planes and cancellation problem2005

    • Author(s)
      K.Masuda, M.Miyanishi
    • Journal Title

      Transactions of the Amer. Math. Soc 357

      Pages: 4867-4883

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Equivariant cancellation for algebraic varieties2005

    • Author(s)
      K.Masuda, M.Miyanishi
    • Journal Title

      Proceedings of the RSME/AMS meeting, Contem.Math.AMS.

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Automorphisms of affine surfaces with $A^1$-fibrations2005

    • Author(s)
      R.V.Gurjar, M.Miyanishi
    • Journal Title

      Michigan Math.J. 53

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Jacobian conjecture and its generalization2005

    • Author(s)
      T.Kambayashi, M.Miyanishi
    • Journal Title

      Proceedings of the RSME/AMS meeting, Contem.Math.AMS

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Nonlinearizable actions of dihedral groups on affine space2004

    • Author(s)
      K.Masuda
    • Journal Title

      Transactions of the Amer. Math. Soc. 356

      Pages: 3545-3556

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Equivariant classfication of Gorenstein open log Pezzo surfaces with finite group actions2004

    • Author(s)
      M.Miyanishi, De-Qi Zhang
    • Journal Title

      J. Math. Soc. Japan 56

      Pages: 215-245

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Nonlinearizable actions of dihedral groups on affine space2004

    • Author(s)
      K.Masuda
    • Journal Title

      Transactions of the Amer.Math.Soc. 356

      Pages: 3545-3556

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Equivariant classification of Gorenstein open log del Pezzo surfaces with finite group actions2004

    • Author(s)
      M.Miyanishi, De-Qi Zhang
    • Journal Title

      J.Math.Soc.Japan 56

      Pages: 215-245

    • NAID

      10013122729

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The additive group actions on Q-homology planes2003

    • Author(s)
      K.Masuda, M.Miyanishi
    • Journal Title

      Ann. Inst. Fourier. Grenoble 53

      Pages: 429-464

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The additive group actions on Q-homology planes2003

    • Author(s)
      K.Masuda, M.Miyanishi
    • Journal Title

      Ann.Inst.Fourier (Grenoble) 53

      Pages: 429-464

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Affine pseudo-planes with torus actions

    • Author(s)
      K.Masuda, M.Miyanishi
    • Journal Title

      Transformation Groups (in press)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Affine lines on affine surfaces and the Maker-Limanov invariant

    • Author(s)
      R.V.Gurjar, K.Masuda et al.
    • Journal Title

      Canadian J. of math. (in press)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Torus actions and kernels of locally nilpotent derivations with slices

    • Author(s)
      K.Masuda
    • Journal Title

      Affine Algebraic Geometry (in honor of Prof. M. Miyanishi) (in press)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A geometric approach to the Jacobian conjecture in dimension two

    • Author(s)
      M.Miyanishi
    • Journal Title

      J. Algebra (in press)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Recent developments in affine algebraic geometry

    • Author(s)
      M.Miyanishi
    • Journal Title

      Affine Algebraic Geometry (in honor of Prof. M. Miyanishi) (in press)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Affine pseudo-planes with torus actions

    • Author(s)
      K.Masuda, M.Miyanishi
    • Journal Title

      Transformation Groups (To appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Affine lines on affine surfaces and the Makar-Limanov invariant

    • Author(s)
      R.V.Gurjar, K.Masuda, M.Miyanishi, P.Russell
    • Journal Title

      Canadian J.of Math. (To appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Torus actions and kernels of locally nilpotent derivations with slices

    • Author(s)
      K.Masuda
    • Journal Title

      "Affine Algebraic Geometry" in honor of Prof.M.Miyanishi (Osaka University Press) (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A geometric approach to the Jacobian conjecture in dimension two

    • Author(s)
      M.Miyanishi
    • Journal Title

      J.Algebra (To appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Recent developments in affine algebraic geometry

    • Author(s)
      M.Miyanishi
    • Journal Title

      "Affine Algebraic Geometry" in honor of Prof.M.Miyanishi (Osaka University Press) (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Affine pseudo-planes with torus actions

    • Author(s)
      K.Masuda, M.Miyanishi
    • Journal Title

      Transformation Groups (in press)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Affine lines on affine surfaces and the Makar-Limanov invariant

    • Author(s)
      R.V.Gurjar, K.Masuda et al.
    • Journal Title

      Canadian J. of Math. (in press)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] A geometric approach to the Jacobian conjecture in dimension two

    • Author(s)
      M.Miyanishi
    • Journal Title

      J. Algebra (in press)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Torus actions and kernels of locally nilpotent derivations with slices

    • Author(s)
      K.Masuda
    • Journal Title

      "Affine Algebraic Geometry" in honor of Prof. M. Miyanishi (in press)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Recent developments in affine algebraic geometry

    • Author(s)
      M.Miyanishi
    • Journal Title

      "Affine Algebraic Geometry" in honor of Prof. M Miyanishi (in press)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Affine pseudo-coverings of affine surfaces

    • Author(s)
      M.Miyanishi
    • Journal Title

      J.Algebra (未定)

    • Related Report
      2004 Annual Research Report
  • [Book] Lectures on Geometry and Topology of Polynomials2004

    • Author(s)
      M.Miyanishi
    • Total Pages
      131
    • Publisher
      Osaka University Press
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Book] Lectures on Geometry and Topology of Polynomials2004

    • Author(s)
      M.Miyanishi
    • Total Pages
      131
    • Publisher
      Osaka University
    • Related Report
      2004 Annual Research Report
  • [Publications] K.Masuda, M.Miyanishi: "The additive group actions on Q-homology planes"Annales de l'Institut Fourier. 53. 429-464 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Masuda: "Nonlinearizable actions of dihedral groups on affine space"Transactions of the American Mathematical Society. (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Miyanishi, De-Qi Zhang: "Equivariant classification of Gorenstein open log Pezzo surfaces with finite group actions"J.Math.Soc.Japan. 56. 215-245 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] R.V.Gurjar, M.Miyanishi: "Automorphisms of affine surfaces with A^1-fibrations"Michigan Math.J.. (2004)

    • Related Report
      2003 Annual Research Report

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

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