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Filtered blowing-up of local rings and algebraic geometric classification of singularities

Research Project

Project/Area Number 16540043
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

TOMARI Masataka  Nihon University, College of Humanities and Sciences, Professor, 文理学部, 教授 (60183878)

Co-Investigator(Kenkyū-buntansha) HAYAKAWA Takayuki  Kanazawa University, Graduate School of Natural Sciences and technology, Lecturer, 大学院・自然科学研究科, 講師 (20198823)
WATANABE Keiichi  Nihon University, College of Humanities and Sciences, Professor, 文理学部, 教授 (10087083)
OKUMA Tomohiro  Yamagata University, Faculty of Education, Art, and Science, Associate Professor, 地域教育文化学部, 助教授 (00300533)
MATSUURA Yutaka  Nihon University, College of Humanities and Sciences, Associate Professor, 文理学部, 助教授 (50096905)
FUKUDA Takuo  Nihon University, College of Humanities and Sciences, Professor, 文理学部, 教授 (00009599)
岩瀬 順一  金沢大学, 大学院・自然科学研究科, 助手 (70183746)
Project Period (FY) 2004 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 2005: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2004: ¥1,800,000 (Direct Cost: ¥1,800,000)
Keywordsresolution of singularities / filtered blowing-up / elliptic singularity / geometric genus / 3-dimensional terminal singularity / F-threshold / 2-dimensional singularity / universal abelican covering / 2次元特異点 / ベロネーゼ部分環 / 完全交叉性 / 2次元楕円型特異点 / 整閉イデアルの対数的特異点解消 / multiplierイデアル / implicit differential systems / 3次元端末的特異点 / discrepancy
Research Abstract

On our research, the following results are given. Some of these were already published, and the rest will be published soon.
(1)The head investigator Tomari studied the normal graded rings whose Veronese subring is a polynomial rings. As an extension of the result in the case of 2-dimensional U.F.D., he obtain the classification of such cases by using famous results of Orlik-Wagreich type. In Tomari's paper, a geometric characterization of complex normal 2-dimensional Gorenstein elliptic singularities was shown by using the recent work of Okuma. Also, in the case of positive characteristic, similar problem was studied and turned out that the techniques in p_g-formula and elliptic singularities can be applied. The lower best bound of the geometric genus of normal 2-dimensional singularities were studied by comparing the topological bound the arithmetic genus. In the graded case, by using non-effective Pinkham-Demazure divisor, the bound for star-shaped singularity is very near the arithm … More etic genus. The studies are still in progress. (2)Hayakawa classified the divisorial contractions to 3-dimensional Gorenstein terminal singularity with small discrepancy. Here he used his special filtered blowing-ups which were introduced in previous his works. In the case of the index with greater than 1, he constructed the terminalization where the exceptional divisor appear with 1/index discrepancy by a composition of irreducible blowing-ups. (3)Watanabe gave a characterization of F-rational ring by using the theory of F-threshold which is a generalization of F-pure threshold. He also gave some interesting inequality of multiplicity in terms of F-threshold. This also induces new results of the case of characteristic zero. (4)Okuma has shown the equisingularity of the universal abelian covers of 2-dimensional rational singularity or minimal elliptic singularity to the complete intersection singularity. This gave a partial answer to the Neuman-Wahl conjecture. He also showed the relation between singularity and its UAC in terms of geometric genus. It turned out that, in the case UAC is of splice type, that the relation is topological. (5)Fukuda obtained the criterion for the existence of solutions for implicit differential systems. He also characterized the singularities which are integrable in the case of the generalized Hamiltonian system, which is important object in the mathematical physics. Less

Report

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

    (23 results)

All 2006 2005 2004 Other

All Journal Article (23 results)

  • [Journal Article] Universal abelian covers of certain surface singularities2006

    • Author(s)
      Tomohiro OKUMA
    • Journal Title

      Math. Ann. 334

      Pages: 753-773

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Universal abelian covers of certain surface singularities2006

    • Author(s)
      T.Okuma
    • Journal Title

      Math.Ann. vol.334

      Pages: 753-773

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Universal abelian covers of certain surface singularities2006

    • Author(s)
      Tomohiro OKUMA
    • Journal Title

      Math.Ann. 334

      Pages: 753-773

    • Related Report
      2005 Annual Research Report
  • [Journal Article] A geometric characterization of normal two-dimensional Gorenstein singularities with $p_a=1$2005

    • Author(s)
      Masataka TOMARI
    • Journal Title

      Proc. The Inst. of Natural Scie. Nihon Univ. 40

      Pages: 179-184

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Gorenstein resolutions of 3-dimensional terminal singularities2005

    • Author(s)
      Takayuki HAYAKAWA
    • Journal Title

      Nagoya Math. J. 178

      Pages: 63-115

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Hilbert-Kunz multiplicity of three-dimensional local rings2005

    • Author(s)
      K.-i.Watanabe, K.-i.Yoshida
    • Journal Title

      Nagoya Math. J. 177

      Pages: 47-75

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Numerical Gorenstein elliptic singularities2005

    • Author(s)
      Tomohiro OKUMA
    • Journal Title

      Math. Zeitschrift 249

      Pages: 31-62

    • 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 characterization of normal two-dimensional Gorenstein singularities with $p_a=1$2005

    • Author(s)
      M.Tomari
    • Journal Title

      Proc.The Inst.of Natural Scie.Nihon Univ. vol.40

      Pages: 179-184

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Gorenstein resolutions of 3-dimensional terminal singularities2005

    • Author(s)
      T.Hayakawa
    • Journal Title

      Nagoya Math.J. vol.178

      Pages: 63-115

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Hilbert-Kunz multiplicity of three-dimensional local rings2005

    • Author(s)
      K.-i.Watanabe, K.-i.Yoshida
    • Journal Title

      Nagoya Math.J. vol.177

      Pages: 47-75

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Numerical Gorenstein elliptic singularities2005

    • Author(s)
      T.Okuma
    • Journal Title

      Math.Zeitschrift vol.249

      Pages: 31-62

    • 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 characterization of normal two-dimensional Gorenstein singularities with $p_a=1$2005

    • Author(s)
      Masataka TOMARI
    • Journal Title

      Proc. The Inst. of Natural Scie.Nihon Univ. 40

      Pages: 179-184

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Divisorial contractions to 3-dimensional terminal singularities with discrepancy one2005

    • Author(s)
      Takayuki HAYAKAWA
    • Journal Title

      J.Math.Soc.Japan 57

      Pages: 653-668

    • NAID

      130000829426

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Gorenstein resolutions of 3-dimensional terminal singularities2005

    • Author(s)
      Tkayuki HAYAKAWA
    • Journal Title

      Nagoya Math.J. 178

      Pages: 63-115

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Numerical Gorenstein elliptic singularities2005

    • Author(s)
      Tomohiro OKUMA
    • Journal Title

      Math.Zeitschrift 249

      Pages: 31-62

    • Related Report
      2005 Annual Research Report
  • [Journal Article] A geometric characterization of normal two-dimensional Gorenstein singularities with $p_a=1$2005

    • Author(s)
      M.Tomari
    • Journal Title

      Proc. The Inst. of Natural Scie. Nihohn Univ. 40(未定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Hilbert-Kunz multiplicity of three-dimensional local rings2005

    • Author(s)
      K.-i.Watanabe, K.-i.Yoshida
    • Journal Title

      Nagoya Math.J. 177

      Pages: 47-75

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Gorenstein resolutions of 3-dimensional tereminal singularities2005

    • Author(s)
      T.Hayakawa
    • Journal Title

      Nagoya Math.J. 178(未定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Singularities of implicit differential systems and their integrability2004

    • Author(s)
      T.Fukuda, S.Janeczko
    • Journal Title

      Banach Center Publications 65

      Pages: 23-47

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Singularities of implicit differential systems and their integrability2004

    • Author(s)
      Fukuda, S.Janeczko
    • Journal Title

      Banach Center Publications vol.65

      Pages: 23-47

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A characterization of semi-quasihomogeneous function in terms of the Milnor number2004

    • Author(s)
      M.Furuya, M.Toamri
    • Journal Title

      Proc.Amer.Math.Soc. 132・7

      Pages: 1885-1890

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Singularities of implicit differential systems and their integrability2004

    • Author(s)
      T.Fukuda, S.janeczko
    • Journal Title

      Banach Center Publications 65

      Pages: 23-47

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Adjacent integrally closed ideals in 2-dimensional regular local rings

    • Author(s)
      S.NOH, Kei-ichi WATANABE
    • Journal Title

      J. of Algebra (to appear)(号・巻、発行年、ページ数未定)

    • Related Report
      2005 Annual Research Report

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Published: 2004-04-01   Modified: 2016-04-21  

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