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Research on singularities of a variety

Research Project

Project/Area Number 14340005
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionTokyo Institute of Technology

Principal Investigator

ISHII Shihoko  Tokyo Institute of Technology, Mathmetics, Professor, 大学院・理工学研究科, 教授 (60202933)

Co-Investigator(Kenkyū-buntansha) FUJITA Takao  Tokyo Institute of Technology, Mathmetics, Professor, 大学院・理工学研究科, 教授 (40092324)
WATANABE Kei-ichi  Nihon University, Mathmetics, Professor, 文理学部, 教授 (10087083)
TOMARI Masataka  Nihon University, Mathmetics, Professor, 文理学部, 教授 (60183878)
TOMARU Tadashi  Gumma University, Medicine, Professor, 医学部, 教授 (70132579)
辻 元  東京工業大学, 大学院・理工学研究科, 助教授 (30172000)
Project Period (FY) 2002 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥12,800,000 (Direct Cost: ¥12,800,000)
Fiscal Year 2005: ¥4,300,000 (Direct Cost: ¥4,300,000)
Fiscal Year 2004: ¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 2003: ¥4,300,000 (Direct Cost: ¥4,300,000)
Fiscal Year 2002: ¥2,200,000 (Direct Cost: ¥2,200,000)
Keywordssingularities / geometric genus / jet scheme / arc space / moduli spase / singularity / toric variety / graded ring / integral closed / complete intersection / multipier ideal / plurigenera / jet scheme
Research Abstract

We gave an affirmative answer to the Nash problem for an arbitrary dimensional toric variety.
We gave a negative answer to the Nash problem in general by showing a counter example of dimension 4.
We studied the structure of the arc space of a toric variety and obtain that the dominating relation of the orbits is translated by the relation of valuations corresponding to the orbits.
We gave an affirmative answer to the Nash problem for non-normal toric variety.
We gave an affirmative answer to the local Nash problem for quasi-ordinary singularities.
We defined a maximal divisorial set on the arc space of a variety and proved that any irreducible component of a contact locus is a maximal divisorial set.
We proved that a maximal divisorial set is represented by the intersection of finite number of contact loci of functions.
We proved that every integrally closed ideal of 2-dimensional regular local ring is a multiplier ideal.
We gave a characterization of 2-dimensional Gorenstein singularities with arithmetic genus 1.

Report

(5 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • Research Products

    (60 results)

All 2005 2004 2003 2002 Other

All Journal Article (46 results) Book (1 results) Publications (13 results)

  • [Journal Article] Arcs, valuations and the Nash maps2005

    • Author(s)
      石井志保子
    • Journal Title

      J. reine angew. Math. 588

      Pages: 71-92

    • 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)
      渡辺敬一, 吉田健一
    • 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] On some classes of weakly Kodaira singularities2005

    • Author(s)
      都丸正
    • Journal Title

      Sem. & Gongres 10

      Pages: 323-340

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] A geometric characterization of normal two-dimensional Gorenstein singularities with pa=12005

    • Author(s)
      泊昌孝
    • Journal Title

      日本大学文理学部自然科学研究所研究紀要 40

      Pages: 179-184

    • NAID

      40006716132

    • 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)
      渡辺敬一
    • Journal Title

      Nagoya Math. J. 177

      Pages: 47-7775

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Arcs, valuations and the Nash map2005

    • Author(s)
      S.Ishii
    • Journal Title

      J.reine angew.Math. 588

      Pages: 71-92

    • 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 rings,2005

    • Author(s)
      K.Watanabe
    • Journal Title

      Nagoya Math.J. 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] A geometric characterization of normal two-dimensional Gorenstein singularities with $p_a=1$,2005

    • Author(s)
      M.Tomari
    • Journal Title

      Proc.The Inst.of Nat Scie.Nihon Univ. 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] On some classes of weakly Kodaira singularities,3402005

    • Author(s)
      T.Tomaru
    • Journal Title

      Singularites Franco-Japonaises

      Pages: 323-323

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Arcs, valuations and the Nash maps2005

    • Author(s)
      石井志保子
    • Journal Title

      J.refine angew.Math. 588

      Pages: 71-92

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Hilbert-Kunz multiplicity of three dimensional local Rings2005

    • Author(s)
      渡辺敬一, 吉田健一
    • Journal Title

      Nagoya Math.J. 177

      Pages: 47-75

    • Related Report
      2005 Annual Research Report
  • [Journal Article] A geometric characterization of normal two-dimensional Gorenstein singularities with pa=12005

    • Author(s)
      泊昌孝
    • Journal Title

      日本大学文埋学部自然科学研究所研究紀要 40

      Pages: 179-184

    • NAID

      40006716132

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Arcs, valuations and the Nash map2005

    • Author(s)
      石井志保子
    • Journal Title

      Journal fur reine und angewandte Mathematik (掲載予定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] A gesmetric characterization of normal 2-dimensinal Gorenstein singularities with P_a=12005

    • Author(s)
      泊昌孝
    • Journal Title

      Proc. Inst.of Natural Sci.Nihon Univ. 40(掲載予定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] The arc space of a toric variety2004

    • Author(s)
      石井志保子
    • Journal Title

      J. Algebra, 278

      Pages: 666-683

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] When does the subadditivity theorem for multiplier ideals hold?2004

    • Author(s)
      渡辺敬一, 高木俊介
    • Journal Title

      Trans. AMS 356

      Pages: 3951-3961

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The total coordinate ring of a normal projective variety,2004

    • Author(s)
      渡辺敬一, Elizondo, 蔵野和彦
    • Journal Title

      J. of Algebra, 276

      Pages: 625-637

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] F-thresholds and Bernstein-Sato polynomials2004

    • Author(s)
      渡辺敬一, Mustata, 高木俊介
    • Journal Title

      4ECM Stockholm

      Pages: 341-363

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] O F-pure thresholds2004

    • Author(s)
      渡辺敬一, 高木俊介
    • Journal Title

      J. Alg. 282

      Pages: 278-297

    • Description
      「研究成果報告書概要(和文)」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)
      泊昌孝, 古屋雅子
    • Journal Title

      Proc. Amer. Math. Soc132, (2004) 1885-189 132

      Pages: 1885-1890

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Extremal functions and prime blow-ups2004

    • Author(s)
      石井志保子
    • Journal Title

      Comm. Algebra 32

      Pages: 819-827

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The arc space of a toric variety2004

    • Author(s)
      S.Ishii
    • Journal Title

      J.Algebra 278

      Pages: 666-683

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Extremal functions and prime blow-ups2004

    • Author(s)
      S.Ishii
    • Journal Title

      Comm.Algebra 32

      Pages: 819-827

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] When does the subadditivity theorem for multiplier ideals hold?2004

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

      Trans.AMS vol.356

      Pages: 3951-3961

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 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(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The total coordinate ring of a normal projective variety,2004

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

      J.of Algebra vol.276, No.2

      Pages: 625-637

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On F-pure thresholds,2004

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

      J.of Algebra 282

      Pages: 278-297

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] F-thresholds and Bernstein-Sato polynomials,2004

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

      4ECM Stockholm

      Pages: 341-363

    • 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.Tomari, M.Furuya
    • Journal Title

      Proc.Amer.Math. Soc132

      Pages: 1885-1890

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Extremal functions and prime blowups2004

    • Author(s)
      石井志保子
    • Journal Title

      Communications in Algebra 32(3)

      Pages: 819-827

    • Related Report
      2004 Annual Research Report
  • [Journal Article] The arc space of a toric variety2004

    • Author(s)
      石井志保子
    • Journal Title

      Journal of Algebra 278

      Pages: 666-683

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

    • Author(s)
      渡辺敬一, 高木俊輔
    • Journal Title

      Trans AMS 356

      Pages: 3951-3961

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

    • Author(s)
      渡辺敬一, 吉田健一
    • Journal Title

      Illinois Journal of Math. 48

      Pages: 625-637

    • Related Report
      2004 Annual Research Report
  • [Journal Article] The Nash problem on arc families of singularities2003

    • Author(s)
      石井志保子, Janos Kollar
    • Journal Title

      Duke Math.J 120

      Pages: 601-620

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Chains of Integrally Closed Ideals2003

    • Author(s)
      渡辺敬一
    • Journal Title

      Contem Math. 331

      Pages: 353-358

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Integrally closed ideals in two-dimensional regular local rings are multiplier ideals2003

    • Author(s)
      渡辺敬一, J.Lipman
    • 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
      2005 Final Research Report Summary
  • [Journal Article] The Nash problem on arc families of singularities2003

    • Author(s)
      S.Ishii, J.Kollar
    • Journal Title

      Duke math.J. 120

      Pages: 601-620

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Chains of Integrally Closed Ideals,2003

    • Author(s)
      K.Watanabe
    • Journal Title

      Contem Math. (AMS) 331

      Pages: 353-358

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Integrally closed ideals in two-dimensional regular local rings are multiplier ideals,2003

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

      Math.Reasearch Letters 10

      Pages: 423-434

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The geometric minimal models of analytic spaces2002

    • Author(s)
      石井志保子, Pierre Milman
    • Journal Title

      Math. Ann. 323

      Pages: 437-451

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Multiplicity of filtere rings and simple K3 singularities of multiplicity two2002

    • Author(s)
      泊昌孝
    • Journal Title

      Publ. RIMS, 38

      Pages: 693-724

    • NAID

      110001050867

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The geometric minimal models of analytic spaces2002

    • Author(s)
      S.Ishii, P.Milman
    • Journal Title

      Math.Ann. 323

      Pages: 437-451

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Multiplicity of filtered rings and simple K3 singularities of mutiplicity two,2002

    • Author(s)
      M.Tomari
    • Journal Title

      Publ.RIMS. 38-4

      Pages: 693-724

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The local Nash problem on arc families of singularities

    • Author(s)
      石井志保子
    • Journal Title

      Annale d'Institut Fourier (印刷中)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The local Nash problem on arc famililes of singularities

    • Author(s)
      S.Ishii
    • Journal Title

      Annale d'Institut Fourier (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] The local Nash problem on arc families of singularities

    • Author(s)
      石井志保子
    • Journal Title

      Annale d'Institut Fourier (掲載予定)(決定)

    • Related Report
      2005 Annual Research Report
  • [Book] 群論の進化2004

    • Author(s)
      堀田, 渡辺, 庄司, 三町
    • Total Pages
      445
    • Publisher
      朝倉書店
    • Related Report
      2004 Annual Research Report
  • [Publications] 石井志保子, Janos Kollar: "The Nash problem on arc families of singularities"Duke Mathematical Journal. 120. 601-620 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 石井志保子: "The arc space of a toric variety"Journal of Algebra. (掲載予定).

    • Related Report
      2003 Annual Research Report
  • [Publications] 渡辺敬一: "Chains do integrally closed ideals"Contemporary Math.(AMS). 331. (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Joseph Lipman, 渡辺敬一: "Integrally closed ideals in two-dimensional regular local rings are multiplier ideals"Math.Research Letters. 10. 423-434 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 高木俊介, 渡辺敬一: "When does the subadditivity theorem for multiplier ideals hold?"Transaction AMS. (掲載予定).

    • Related Report
      2003 Annual Research Report
  • [Publications] 都丸 正: "On some classes of weakly Kodaira singularties"Proc.of Franco-Japonais Luminy Conference of Singularities. (掲載予定).

    • Related Report
      2003 Annual Research Report
  • [Publications] 石井志保子: "The invariant-K^2 and continued fraction for 2-dimensional cycle quotest singhanties"Abhandlungen Math. Sem. Hamburg. 72. 207-215 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 石井志保子: "Extremal functions and prime blow ups"Comm. Alg.. (掲載予定).

    • Related Report
      2002 Annual Research Report
  • [Publications] 石井志保子, Pierre Milman: "The geometric minimal model of analytie spaces"Math. Ann.. 323. 437-451 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 石井志保子, Jamos Kollar: "The Nash problem on arc families of singularities"Duke Math. J.. (掲載予定).

    • Related Report
      2002 Annual Research Report
  • [Publications] 渡辺敬一, 原伸生, 吉田健一: "F-rationality of Rees algebra"J of Algebra. 247. 153-190 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 辻本, G.Schumacher: "Quasi projectivity of Moduli space of polarized varieties"Annulus of Math. (掲載予定).

    • Related Report
      2002 Annual Research Report
  • [Publications] 渡辺 敬一: "環と体"朝倉書店. 179 (2002)

    • Related Report
      2002 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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