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Explicit studies on singularities which appear on 3-dimensional algebraic varieties

Research Project

Project/Area Number 17540019
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

HAYAKAWA Takayuki  Graduate School of Natural Science & Technology, Lecturer, 自然科学研究科, 講師 (20198823)

Co-Investigator(Kenkyū-buntansha) SUGANO Takashi  Graduate School of Natural Science & Technology, Professor, 自然科学研究科, 教授 (30183841)
MORISHITA Masanori  Kyushu University, Faculty of Mathematics, Professor, 数理学研究院, 教授 (40242515)
IWASE Zjunici  Graduate School of Natural Science & Technology, Assistant, 自然科学研究科, 助手 (70183746)
Project Period (FY) 2005 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2006: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2005: ¥900,000 (Direct Cost: ¥900,000)
Keywordsterminal singularity / divisorial contraction / resolution of singularity
Research Abstract

We studied on resolutions of 3-dimensional terminal singularities and 3-dimensional divisorial contractions and get the results below.
(1) We construct birational morphisms from 3-folds with only terminal singularities with index 1 to 3-dimensional terminal singularities with index m greater than one, by repeating divisorial contractions with minimal discrepancy. As a result we determined all prime divisors over 3-dimensional terminal singularities with index greater than or equal to 2 which have discrepancy not greater than 1.
(2) We showed that there does not exist economic resolutions due to M.Reid in its original form, which states that if there are partial resolutions of 3-dimensional terminal singularities with all prime divisors with discrepancy not greater than one as its exceptional divisors. We also showed that there are such economic resolutions after changing conditions on singularities.
(3) We gave an explicit description of divisorial contractions which contract divisors to points with discrepancy one, for 3-dimensional terminal singularities with index greater than one. Such divisorial contractions are obtained by embedding singularities into four or five dimensional spaces and by making suitable weighted blowing ups.
(4) We gave an explicit description of divisorial contractions of Gorenstein terminal singularities of type cD and cE with discrepancy one.

Report

(3 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report
  • Research Products

    (15 results)

All 2006 2005

All Journal Article (15 results)

  • [Journal Article] Pro-p link groups and p-homology groups2006

    • Author(s)
      J.Hillman, D.Matei, M.Morishita
    • Journal Title

      Contemp. Math. 416

      Pages: 121-136

    • 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 Alexander stratification in the deformation space of Galois characters2006

    • Author(s)
      Masanori Morishita
    • Journal Title

      Kyushu J. Math. 60・2

      Pages: 405-414

    • NAID

      130000063148

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Morishita, Pro-p link groups and p-homology groups2006

    • Author(s)
      J.Hillman, D.Matei, M.
    • Journal Title

      Contemp. Math. 416

      Pages: 121-135

    • 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 Alexander stratification in the deformation space of Galois characters2006

    • Author(s)
      M.Morishita
    • Journal Title

      Kyushu J., Math. 60・2

      Pages: 405-414

    • NAID

      130000063148

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Pro-p link groups and p-homology groups2006

    • Author(s)
      Hillman, J., Matei, D., Morishita, M.
    • Journal Title

      Contemp. Math. 416

      Pages: 121-136

    • Related Report
      2006 Annual Research Report
  • [Journal Article] On the Alexander stratification in the deformation space of Galois characters2006

    • Author(s)
      Morishita, Masanori
    • Journal Title

      Kyushu J. Math. 60.2

      Pages: 405-414

    • NAID

      130000063148

    • Related Report
      2006 Annual Research Report
  • [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
      2006 Final Research Report Summary
  • [Journal Article] A remark on partial resolutions of 3-dimensional terminal singularities2005

    • Author(s)
      Takayuki Hayakawa
    • Journal Title

      Nagoya Math. J. 178

      Pages: 117-127

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Divisorial contractions to 3-dimensional terminal singularities with discrepancy one2005

    • Author(s)
      Takayuki Hayakawa
    • Journal Title

      J. Math. Soc. Japan 57

      Pages: 651-668

    • NAID

      130000829426

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Gorenstein resolutions of 3-dimensional terminal singularities2005

    • Author(s)
      T.Hayakawa
    • Journal Title

      Nagoya Math. J. 178

      Pages: 63-115

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] A remark on partial resolutions of 3-dimensional terminal singularities2005

    • Author(s)
      T.Hayakawa
    • Journal Title

      Nagoya Math. J. 178

      Pages: 117-127

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Divisorial contractions to 3-dimensional terminal singularities with discrepancy one2005

    • Author(s)
      T.Hayakawa
    • Journal Title

      J. Math. Soc. Japan 57

      Pages: 651-668

    • NAID

      130000829426

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Gorenstein resolutions of 3-dimensional terminal singularities2005

    • Author(s)
      Takayuki Hayakawa
    • Journal Title

      Nagoya Mathematical Journal 178

      Pages: 63-115

    • Related Report
      2005 Annual Research Report
  • [Journal Article] A remark on partial resolutions of 3-dimensional terminal Singularities2005

    • Author(s)
      Takayuki Hayakawa
    • Journal Title

      Nagoya Mathematical Journal 178

      Pages: 117-127

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

    • Author(s)
      Takayuki Hayakawa
    • Journal Title

      Journal of the Mathematical Society of Japan 57

      Pages: 651-668

    • NAID

      130000829426

    • Related Report
      2005 Annual Research Report

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Published: 2005-04-01   Modified: 2016-04-21  

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