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Commutative Ring Theory of Singularities of Algebraic Varieties

Research Project

Project/Area Number 23540059
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionNihon University

Principal Investigator

WATANABE Kei-ichi  日本大学, 文理学部, 教授 (10087083)

Co-Investigator(Kenkyū-buntansha) YOSHIDA Ken-ichi  日本大学, 文理学部, 教授 (80240802)
Co-Investigator(Renkei-kenkyūsha) KURANO Kazuhiko  明治大学, 理工学部, 教授 (90205188)
TAKAGI Shunsuke  東京大学, 大学院・数理科学研究科, 准教授 (40380670)
Project Period (FY) 2011 – 2013
Project Status Completed (Fiscal Year 2013)
Budget Amount *help
¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000)
Fiscal Year 2013: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Fiscal Year 2012: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2011: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords環論 / 代数学 / 特異点 / 可換環論 / 有理特異点 / 重複度 / F-threshold / F-pure rings / Hilbert-Kunz 重複度 / a 不変量 / F-pure / F-rational / jet scheme / 一般化された test ideal / 半群環
Research Abstract

The aim of this research is to apply commutative ring theory to the study of singular points of algebraic varieties and, conversely, to express properties of commutative rings in terms of geometric language of singularities and analyze it.
Main results of this research are the followings; 1. Obtained upper bound of multiplicities of f-pure rings in terms of dimension and embedding dimension, 2. Described the quasi-socle ideals of parameter ideals using the tight closure theory of ideals of positive characteristic and got a bound, 3. Obtained the complete classification of Ulrich ideals in 2-dimensional rational singularities, 4. Characterized almost Gorenstein semigroup rings in affine 3-space in terms of its free resolution, 5. Obtained inequalities between a-invariant and F-pure threshold of homogeneous toric rings, 6. Determined when the jet schemes of a hypersurface have rational singularities.

Report

(4 results)
  • 2013 Annual Research Report   Final Research Report ( PDF )
  • 2012 Research-status Report
  • 2011 Research-status Report
  • Research Products

    (37 results)

All 2014 2013 2012 2011 Other

All Journal Article (19 results) (of which Peer Reviewed: 19 results) Presentation (17 results) (of which Invited: 2 results) Book (1 results)

  • [Journal Article] F特異点--正標数の手法の特異点論への応用2014

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

      「数学」(日本数学会)

      Volume: 66巻 Pages: 1-30

    • NAID

      130006882583

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] Ulrich ideals and modules2014

    • Author(s)
      S. Goto, Kazuho Ozeki, , Ryo Takahashi, Kei-ichi Watanabe, Ken-ichi . Yoshida
    • Journal Title

      Mathematical Proceedings of the Cambridge Philosophical Society

      Volume: vol.156 Pages: 137-166

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] F-thresholds vs. a-invariants for homogeneous toric rings2014

    • Author(s)
      Daisuke Hirose, Kei-ichi Watanabe, Ken-ichi . Yoshida
    • Journal Title

      Communications in Algebra

      Volume: vol 42 Pages: 2704-2720

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] F-thresholds vs. a-invariants for homogeneous toric rings2014

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

      Communications in Algebra

      Volume: 42 Pages: 2704-2720

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Ulrich ideals and modules2014

    • Author(s)
      S. Goto, K. Ozeki, R. Takahashi, K. Watanabe, K. Yoshida
    • Journal Title

      Mathematical Proceedings of the Cambridge Philosophical Society

      Volume: 156 Pages: 137-166

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Upper bound of multiplicity of F-rational rings and F-pure rings,2014

    • Author(s)
      C. Huneke, K. Watanabe
    • Journal Title

      Proc. Amer. Math. Soc.

      Volume: To appear

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A variant of Wang's theorem2012

    • Author(s)
      Kei-ichi Watanabe, Ken-ichi . Yoshida
    • Journal Title

      J. of Algebra

      Volume: vol.369 Pages: 129-145

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] Jet schemes of homogeneous hypersurfaces, "Singularities in Geometry and Topology, Strasbourg 2009"2012

    • Author(s)
      Shihoko Ishii, Akiyoshi Sannai, Kei-ichi Watanabe
    • Journal Title

      IRMA Lectures in Mathematics and Theoretical Physics 20, European Mathematical Society

      Pages: 39-49

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] Genus of numerical semigroups generated by three elements2012

    • Author(s)
      Hirokatsu Nari, Takahiro Numata, Kei-ichi Watanabe
    • Journal Title

      J. of Algebra

      Volume: vol.358 Pages: 67-73

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] Genus of numerical semigroups generated by three elements,2012

    • Author(s)
      Hirokatsu Nari, Takahiro Numata, Kei-ichi Watanabe
    • Journal Title

      Journal of Algebra

      Volume: 358 Pages: 67-73

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Jet schemes of homogeneous hypersurfaces,2012

    • Author(s)
      Shihoko Ishii, Akiyoshi Sannnai, Kei-ichi Watanabe
    • Journal Title

      IRMA Lectures in Mathematics and Theoretical Physics , European Mathematical Society

      Volume: 20 Pages: 39-49

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] A variant of Wang's theorem,2012

    • Author(s)
      Kei-ichi Watanabe, Ken-ichi Yoshida
    • Journal Title

      Journal of Algebra

      Volume: 369 Pages: 129-145

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Genus of numerical semigroups generated by three elements,2012

    • Author(s)
      H. Nari, T. Numata, K. Watanabe
    • Journal Title

      Jouranal of Algebra

      Volume: 358 Pages: 67-73

    • Related Report
      2011 Research-status Report
    • Peer Reviewed
  • [Journal Article] Multiplicity bounds in graded rings2011

    • Author(s)
      Craig Huneke, S. Takagi, Kei-ichi Watanabe
    • Journal Title

      Kyoto Journal of Mathematics

      Volume: vol.51 Pages: 127-147

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] F-signature of graded Gorenstein rings2011

    • Author(s)
      Akiyoshi Sannai, Kei-ichi Watanabe
    • Journal Title

      J. Pure and Appl. Algebra

      Volume: vol.215 Pages: 2190-2195

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] Coefficient ideal of ideals generated by monomials2011

    • Author(s)
      Satoshi Ohnishi, Kei-ichi Watanabe
    • Journal Title

      Communications in Algebra

      Volume: vol.39 Pages: 1563-1576

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] Multiplicity bounds in graded rings, Kyoto Journal of Mathematics2011

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

      Kyoto Journal of Mathematics

      Volume: 51 Pages: 127-147

    • Related Report
      2011 Research-status Report
    • Peer Reviewed
  • [Journal Article] Classification of 2-dimensional normal graded hypersurfaces with a(R) ¥le 6

    • Author(s)
      Kei-ichi Watanabe
    • Journal Title

      Proceedings of the 2012 Conference at IMPA

      Volume: (To appear)

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] Upper bound of multiplicity of F-rational rings and F-pure rings

    • Author(s)
      Craig Huneke, . Kei-ichi Watanabe
    • Journal Title

      Proc. American Mathematical Soc

      Volume: (To appear)

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Presentation] Good ideals of 2-dimensional normal singularities2013

    • Author(s)
      Kei-ichi Watanabe
    • Organizer
      International Conference on Commutative Algebra and its Interaction to Algebraic Geometry and Combinatorics
    • Place of Presentation
      Institute of Math., Hanoi
    • Year and Date
      2013-12-09
    • Related Report
      2013 Final Research Report
  • [Presentation] Good ideals of 2-dimensional normal singularities2013

    • Author(s)
      Kei-ichi Watanabe
    • Organizer
      COMMUTATIVE ALGEBRA AND ITS INTERACTION WITH ALGEBRAIC GEOMETRY
    • Place of Presentation
      CIRM, France
    • Year and Date
      2013-07-10
    • Related Report
      2013 Final Research Report
  • [Presentation] Existence of good ideals in 2-dimensional normal Gorenstein rings2013

    • Author(s)
      Kei-ichi Watanabe
    • Organizer
      Commutative Algebra Seminar
    • Place of Presentation
      MSRI, UC Berkeley
    • Year and Date
      2013-03-14
    • Related Report
      2013 Final Research Report
  • [Presentation] Ulrich modules and Special modules over 2-dimensional rational singularities2012

    • Author(s)
      吉田健一・渡辺敬一
    • Organizer
      第45回環論および表現論シンポジウム
    • Place of Presentation
      信州大学
    • Year and Date
      2012-09-08
    • Related Report
      2013 Final Research Report
  • [Presentation] On Ulrich ideals and modules, II2012

    • Author(s)
      Kei-ichi Watanabe
    • Organizer
      12th ALGA meeting
    • Place of Presentation
      IMPA, Rio de Janeiro
    • Year and Date
      2012-08-16
    • Related Report
      2013 Final Research Report
  • [Presentation] Ulrich modules and Special modules over 2-dimensional rational singularities,2012

    • Author(s)
      吉田健一・渡辺敬一
    • Organizer
      第45回環論および表現論シンポジウム
    • Place of Presentation
      信州大学
    • Related Report
      2012 Research-status Report
  • [Presentation] The upper bound of multiplicity of F-pure rings and rational singularities2011

    • Author(s)
      Kei-ichi Watanabe
    • Organizer
      The 7th Japan- Vietnam Joint Seminar on Commutative Algebra
    • Place of Presentation
      Quy Nhon University, Vietnam
    • Year and Date
      2011-12-13
    • Related Report
      2013 Final Research Report
  • [Presentation] Upper bound of multiplicity of F-rational and F-pure rings2011

    • Author(s)
      渡辺 敬一
    • Organizer
      第33回可換環論シンポジウム
    • Place of Presentation
      浜名湖カリアック
    • Year and Date
      2011-11-09
    • Related Report
      2013 Final Research Report
  • [Presentation] 11/8 トーリック環上の F-thresholds2011

    • Author(s)
      吉田 健一, 渡辺 敬一
    • Organizer
      第33回可換環論シンポジウム
    • Place of Presentation
      浜名湖カリアック
    • Year and Date
      2011-11-08
    • Related Report
      2013 Final Research Report
  • [Presentation] Certain Jet Schemes have Rational Singularities2011

    • Author(s)
      渡辺 敬一
    • Organizer
      特異点論とそのひろがり
    • Place of Presentation
      京都大学
    • Year and Date
      2011-08-24
    • Related Report
      2013 Final Research Report
  • [Presentation] Certain Jet Schemes have Rational Singularities2011

    • Author(s)
      Kei-ichi Watanabe
    • Organizer
      Midwest Commutative Algebra and Geometry Conference
    • Place of Presentation
      Purdue University
    • Year and Date
      2011-05-15
    • Related Report
      2013 Final Research Report
  • [Presentation] Certain Jet Schemes have Rational Singularities2011

    • Author(s)
      K. Watanabe
    • Organizer
      Midwest Commutative Algebra and Geometry Conference,(招待講演)
    • Place of Presentation
      Purdue University, USA
    • Related Report
      2011 Research-status Report
  • [Presentation] The upper bound of multiplicity of F-pure rings and rational singularities2011

    • Author(s)
      K. Watanabe
    • Organizer
      The 7th Japan- Vietnam Joint Seminar on Commutative Algebra(招待講演)
    • Place of Presentation
      Quy Nhon University, Vietnam
    • Related Report
      2011 Research-status Report
  • [Presentation] Good ideals of 2-dimensional normal singularities,

    • Author(s)
      K. Watanabe
    • Organizer
      COMMUTATIVE ALGEBRA AND ITS INTERACTION WITH ALGEBRAIC GEOMETRY,
    • Place of Presentation
      CIRM, France
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Good ideals on 2-dimensional normal singularities,

    • Author(s)
      T. Okuma, K. Watanabe, K. Yoshida
    • Organizer
      35-th Commutative Algebra Symposium,
    • Place of Presentation
      京都大学数理解析研究所
    • Related Report
      2013 Annual Research Report
  • [Presentation] On Ulrich ideals and modules, II,

    • Author(s)
      Kei-ichi Watanabe
    • Organizer
      12th ALGA meeting,
    • Place of Presentation
      IMPA, Rio de Janeiro,
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] Existence of good ideals in 2-dimensional normal Gorenstein rings

    • Author(s)
      Kei-ichi Watanabe
    • Organizer
      Commutative Algebra Seminar,
    • Place of Presentation
      MSRI, UC Berkeley,
    • Related Report
      2012 Research-status Report
  • [Book] 可換環論2011

    • Author(s)
      後藤四郎,渡辺敬一
    • Total Pages
      350
    • Publisher
      日本評論社
    • Related Report
      2011 Research-status Report

URL: 

Published: 2011-08-05   Modified: 2019-07-29  

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