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2001 Fiscal Year Final Research Report Summary

Syzygies for the defining ideal of projective varieties

Research Project

Project/Area Number 11640052
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionUniversity of the Ryukyus

Principal Investigator

MIYAZAKI Chikashi  University of the Ryukyus College of Science, Associate Professor, 理学部, 助教授 (90229831)

Co-Investigator(Kenkyū-buntansha) FUJISAWA Taro  Nagano National College of Technology, Associate Professor, 一般科, 助教授 (60280385)
SUGA Shuichi  University of the Ryukyus College of Science, Associate Professor, 理学部, 助教授 (30206388)
MAEDA Takashi  University of the Ryukyus College of Science, Professor, 理学部, 教授 (30229306)
Project Period (FY) 1999 – 2001
KeywordsCastelnuovo / free resolition / syzygy / algebraic curve
Research Abstract

My research has been devoted to the study of free resolutions for defining ideals of projective varieties, especially to that of the Castelnuovo-Mumford regularities. The regularity is a basic invatiant which describes the minimal free resolutions and the degrees of the defining equations of the varieties. Let X ⊂ P^N_K be a projective variety over an algebraically closed field K. Then, by using an invariant k(X) which evaluates the deficiency of the Hartshorne-Rao module of the variety, we have known an upper bound on the regularity reg(X) 【less than or equal】[(deg(X)-1)/codim(X)] + max { k(X)・dim(X), 1}. In order to classify the equality case, we consider a generic hyperplane section of the projective curve satisfying reg(X) = [(deg(X)-1)/codim(X)] + 1. In case char(k) = 0, the uniform position principle yields an information on the configuration of the zero-dimensional scheme, and the set of points as a generic hyperplane section is contained in a rational normal curve. In case char(k) > 0, the correspodence between the monodromy group of the projective curve and the configuration of the points excludes the strange curves. Thus, by dimensional induction, the sharp bounds are only appeared in the case of divisors on a Hirzeburch surfaces if X is not arithmetically Cohen-Macaulay. Further I have conjectured with Le Tuan Hoa, by introducing a new invariant k^^〜(X), reg(X) 【less than or equal】[(deg(X) -1)/codim(X)] + max.{ k^^〜(X), 1}, and the bound is obtained to be effective for the divisor on the rational normal scroll.

  • Research Products

    (20 results)

All Other

All Publications (20 results)

  • [Publications] E.Ballico, C.Miyazaki: "Generic hyperplane section of curves and an application to regularity bounds in positive characteristic"J. Pure Appl. Algebra. 155. 93-103 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] C.Miyazaki: "Sharp bounds on Castelnuovo-Mumford regularity"Trans. Amer. Math. Soc.. 352. 1675-1686 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] C.Miyazaki, W.Vogel: "Arithemic and geometric degrees of graded modules"Communitative algebra, algebraic geomethy and computational methods (ed. Eisenbud). 97-112 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Maeda: "Minimal algebra resolution associated with hook representations"J. Algebra. 237. 287-291 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Maeda: "Determinantal equations and singular loci of duals of Grassmanions"Ryukyu Math. J.. 14. 17-40 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Maeda: "Generic G-linear maps"Ryukyu Math. J.. 13. 23-46 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Maeda: "Standard P^3 -bundles of exponent two on algebraic surfaces"Comm. Algebra. 28. 2858-2868 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Maeda: "Minimal free resolution of the Hurd Veronese subring of three variables"Ryukyu Math. J.. 12. 9-30 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Fujisawa: "Degeneration of weight spectral sequences"Manuscripta Math.. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Fujisawa: "Linuits of Hodge structures in several variables"Compositio Math.. 114. 129-183 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Edoardo Ballico, Chikashi Miyazaki: "Generic hyperplane section of curves and an application to regularity bounds in positive characteristic"J. Pure Appl. Algebra. 155. 93-103 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Chikashi Miyazaki: "Sharp bounds on Castelnuovo-Mumford regularity"Trans. Amer. Math. Soc.. 352. 1675-1686 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Chikashi Miyazaki, Wolfgang Vogel: "Arithmetic and geometric degrees of graded modules, Commutative algebra, algebraic geometry, and computational methods (ed. D. Eisenbud)"Springer. 97-112 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Takashi Maeda: "Minimal algebra resolution associated with hook representations"J. Algebra. 237. 287-291 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Takashi Maeda: "Determinantal equations and singular loci of duals of Grassmannians"Ryukyu Math. J. 14. 17-40 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Takashi Maeda: "Generic G-linear maps"Ryukyu Math. J.13. 23-46 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Takashi Maeda: "Standard P^3-bundles of exponent two on algebraic surfaces"Comm. Algebra. 28. 2853-2868 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Takashi Maeda: "Minimal free resolution of the third Veronese subring of three variables"Ryukyu Math.. J12. 9-30 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Taro Fujisawa: "Degeneration of weight spectral sequences"(to appear) in Manuscripta math..

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Taro Fujisawa: "Limits of Hodge structures in several variables"Compositio Math.. 114. 129-183 (1999)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2003-09-17  

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