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Commutative ring theory and singularity theory

Research Project

Project/Area Number 13440015
Research Category

Grant-in-Aid for Scientific Research (B)

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

Principal Investigator

WATANABE Keiichi  Nihon University, College of Humanities and Sciences, Professor, 文理学部, 教授 (10087083)

Co-Investigator(Kenkyū-buntansha) TOMARI Masataka  Nihon University, College of Humanities and Sciences, Professor, 文理学部, 教授 (60183878)
FUKUDA Takuo  Nihon University, College of Humanities and Sciences, Professor, 文理学部, 教授 (00009599)
MOTEGI Kamahi  Nihon University, College of Humanities and Sciences, Professor, 文理学部, 教授 (40219978)
KURANO Kazuhiko  Meiji University, School of Science and Technology, Professor, 理工学部, 教授 (90205188)
HARA Nobuo  Tohoku University, Graduate School of Science, Assistant Lecturer, 大学院・理学研究科, 助教授 (90298167)
鈴木 正彦  日本大学, 文理学部, 教授 (00171249)
Project Period (FY) 2001 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥7,100,000 (Direct Cost: ¥7,100,000)
Fiscal Year 2004: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 2003: ¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 2002: ¥1,900,000 (Direct Cost: ¥1,900,000)
Fiscal Year 2001: ¥2,100,000 (Direct Cost: ¥2,100,000)
Keywordsmultiplier ideal / lc threshold / F-pure / Hilbert-Kunz multiplicity / Frobenius endomorphism / log terminal singularity / tight closure / regular local ring / F-pure threshold / 整閉イデアル / log resolution / UFD / Seifelt手術 / Chow群 / 特異点 / 可換環論 / 全座標環 / Hilbert-Kunz重複度 / 素元分解環 / 特異点の解消 / tight closure / 有理特異点 / subadditivity / regular ring / F-regular ring / terminal singularity / blowing up / 商特異点 / F-pure ring
Research Abstract

The results are mainly concerning the followings 3 themes
1.Multiplier ideals ;
J.Lipman and K.Watanabe proved that every integrally closed ideal in 2 dimensional regular local rings is a multiplier ideal.
N.Hara and K.Yoshida defined a generalization of "tight closures" in characteristic p>0 and by using that concept, they Succeeded to calculate multiplier ideals by purely algebraic (by commutative ring theory) method.
S.Takagi and K.Watanabe established the notion of "F-pure thresholds", which corresponds to the notion of lc(=log canonical) threshold in characteristic 0, used in algebraic geometry. This concept has many interesting features in both singularity theory and commutative ring theory.
2.Hilbert-Kunz multiplicity ;
Hilbert-Kunz multiplicity is a kind of multiplicity defined for rings of positive characteristics. Watanabe and Yoshida proved before that a ring is regular if and only if the HK multiplicity of the ring is 1. This time we determined the rings whose HK multiplicity is smallest among non-regular rings in dimension 2 and 3.

Report

(5 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (40 results)

All 2005 2004 2003 2002 2001 Other

All Journal Article (20 results) Book (1 results) Publications (19 results)

  • [Journal Article] Hilbert-Kunz multiplicity of three dimensional local rings2005

    • Author(s)
      K.Watanabe, K.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
      2004 Final Research Report Summary
  • [Journal Article] On F-pure thresholds,2004

    • Author(s)
      S.Takagi, K.Watanabe
    • Journal Title

      Journal of Algebra, 282

      Pages: 278-297

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Minimal relative Hilbert-Kunz multiplicity2004

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

      Illinois J.Math. 48

      Pages: 273-294

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The total coordinate ring of a normal projective variety,2004

    • Author(s)
      J.Elizondo, K.Kurano, K.Watanabe
    • Journal Title

      Journal of Algebra, 276

      Pages: 625-637

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The total coordinate ring of a normal projective variety2004

    • Author(s)
      J.Elizondo, K.Kurano, K.Watanabe
    • Journal Title

      J.of Algebra vol.276

      Pages: 625-637

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On F-pure thresholds2004

    • Author(s)
      S.Takagi, K.Watanabe
    • Journal Title

      J.of Algebra vol.282

      Pages: 278-297

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A characterization of semi-quasihomogeneous Function in terms of the Milnor number2004

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

      Proc.of Amer.Math.Soc. vol.132

      Pages: 1885-1890

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The total coordinate ring of normal projective variety2004

    • Author(s)
      J.Elizondo, K.Kurano, K.Watanabe
    • Journal Title

      Journal of Algebra 276

      Pages: 625-637

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On F-pure thresholds2004

    • Author(s)
      S.Takagi, K.Watanabe
    • Journal Title

      J.of Algebra 282

      Pages: 278-297

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Minimal relative Hilbert-Kunz multiplicity2004

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

      Illinois J.of Mathematics 48

      Pages: 273-294

    • Related Report
      2004 Annual Research Report
  • [Journal Article] When does the subadditivity theorem for multiplier ideals hold?2004

    • Author(s)
      S.Takagi, K.Watanabe
    • Journal Title

      Transaction of American Mathematical Society 356

      Pages: 3951-3961

    • Related Report
      2004 Annual Research Report
  • [Journal Article] A characterization of semi-quasihomogeneous functions in terms of Milnor numbers2004

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

      Proceedings of American Mathematical Society 132

      Pages: 1885-1890

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On a generalization of test ideals2004

    • Author(s)
      N.Hara, S.Takagi
    • Journal Title

      Nagoya Mathematical Journal 175

      Pages: 59-74

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Integrally closed ideals in two-dimensional regular local rings are multiplier ideals,2003

    • Author(s)
      J.Lipman, K.Watanabe
    • Journal Title

      Math.Reasearch Letters 10

      Pages: 423-434

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A generalization of tight closure and multiplier ideals 355 (2003), 3143--3174.2003

    • Author(s)
      N.Hara, K.Yoshida
    • Journal Title

      Transaction of American Math.Soc. 355

      Pages: 3143-3174

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Integrally closed ideals in two-dimensional regular local rings are multiplier ideals2003

    • Author(s)
      J.Lipman, K.Watanabe
    • Journal Title

      Math.Reasearch Letters vol.10

      Pages: 423-434

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A generalization of tight closure and multiplier ideals2003

    • Author(s)
      N.Hara, K.Yoshuda
    • Journal Title

      Trans.Amer.Math.Soc. vol 355

      Pages: 3143-3174

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] F-regular and F-pure rings vs.log terminal and log canonical singularities.2002

    • Author(s)
      N.Hara, K.Watanabe
    • Journal Title

      Journal of Algebraic Geometry vol 11

      Pages: 363-392

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] F-rationality of Rees algebras2002

    • Author(s)
      N.Hara, K.Watanabe, K.Yoshida
    • Journal Title

      Journal of Algebra vol 247

      Pages: 153-190

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On Roberts rings,2001

    • Author(s)
      K.Kurano
    • Journal Title

      J.of Mathematical Society of Japan 53

      Pages: 333-355

    • NAID

      10006654637

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Book] 環と体2002

    • Author(s)
      渡辺 敬一
    • Total Pages
      180
    • Publisher
      朝倉書店
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Publications] J.Elizondo, K.Kurano, K.Watanabe: "The total coordinate ring of a normal projective variety"J.of Algebra. (to appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Watanabe, K.Yoshida: "Minimal relative Hilbert-Kunz multiplicity"Illinois J.Math.48(2004)273-294(with K.Yoshida). 48. 273-294 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Furuya, M.Tomari: "A characterization of semi-quasihomogeneous function in terms of the Milnor number"Proc.Amer.Math.Soc.. (to appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Kurano: "Numerical equivalence defined on Chow groups of Noetherian local rings"Invent.Math.. (to appera).

    • Related Report
      2003 Annual Research Report
  • [Publications] N.Hara, K.Yoshida: "A generalization of tight closure and multiplier ideals"Trans.Amer.Math.Soc.. 355. 3143-3174 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] A.D.Matignon, K.Motegi: "Twisted unknots"C.R.Acad.Sci.Paris, Ser.I. 337. 321-326 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] N.Hara, K.Watanabe: "F-regular and F-pure rings vs. log terminal and log canonical singularities"J. of Algebraic Geometry. 11. 363-392 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] N.Hara, K.Watanabe, K.Yoshida: "Rees Algebras of F-regular type"J. of Algebra. 247. 191-218 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Fukuda, S.Janecko: "Smooth integrability of implicit differential equations"Banach center Publication, Polish Acad. of Sci.. (To aAppear).

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Miyazaki, K.Motegi: "Crossing change and exceptional Dehn surgery"Osaka J. Math.. 40. 1-5 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Tomari: "Multiplicity of filtered rings and simple K3 singularities of multiplicity two"Publ. Rims, Kyoto Univ.. 38-4. 693-724 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Kurano, A.Singa: "Todd classes of affine cones of Grassmannians"Int. Math. Res. Notices,. 35. 1841-1855 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 渡辺敬一: "環と体"朝倉書店. 184 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Tomari, K.Watanabe: "Cyclic Covers of Normal Graded Rings"Kodai Math.J.. 24. 436-457 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] N.Hara, K.Watanabe, K.Yoshida: "F-rationality of Rees algebras"j.of Algebra. 247. 153-190 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] N.Hara, K.Watanabe, K.Yoshida: "Rees algebras of F-regular type"j.of Algebra. 247. 191-218 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] N.Hara: "Geometric interpretation of tight closure and test ideals"Trans.Amer.Math.Soc.. 353. 1885-1906 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Tomari: "Multiplicity of filtered rings and simple K3 singularities of multiplicity two"Publ.RIMS Kyoto Univ.. (To appear)(accepted).

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Kurano: "On Roberts rings"J.Math.Soc.Japan. 53. 333-355 (2001)

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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