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Understanding the structure of open algebraic surfaces and normal algebraic surfaces of logarithmic Kodaira dimension one or less

Research Project

Project/Area Number 26400042
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

Kojima Hideo  新潟大学, 自然科学系, 教授 (90332824)

Co-Investigator(Renkei-kenkyūsha) KISHIMOTO Takashi  埼玉大学, 理工学研究科, 准教授 (20372576)
SAITO Natsuo  広島市立大学, 情報科学研究科, 講師 (70382372)
TAKAHASHI Takeshi  新潟大学, 自然科学系, 准教授 (60390431)
Research Collaborator NAGAMINE Takanori  
Project Period (FY) 2014-04-01 – 2017-03-31
Project Status Completed (Fiscal Year 2016)
Budget Amount *help
¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000)
Fiscal Year 2016: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2015: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2014: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Keywords代数幾何学 / 開代数曲面 / 正規代数曲面 / 対数的小平次元 / 多項式環 / 高階導分 / 正規デルペッゾ曲面 / Qホモロジー射影平面 / 消去問題
Outline of Final Research Achievements

I have studied open algebraic surfaces, normal algebraic surfaces and kernels of higher derivations in polynomial rings. I proved that, for an irrational open algebraic surface, its logarithmic Kodaira dimension is non-negative if and only if its logarithmic 12 genus is positive. I studied normal del Pezzo surfaces of Picard rank one with only rational log canonical singularities by using structure theorems on open algebraic surfaces and some results on Q-homology planes and gave partial classification results for those surfaces with one or four singular points. I also gave a sufficient condition for the kernel of a locally finite higher derivations in the polynomial ring in three variables to be a polynomial ring. Furthermore, I applied these results to some problems on affine algebraic varieties.

Report

(4 results)
  • 2016 Annual Research Report   Final Research Report ( PDF )
  • 2015 Research-status Report
  • 2014 Research-status Report
  • Research Products

    (18 results)

All 2017 2016 2015 2014

All Journal Article (4 results) (of which Peer Reviewed: 4 results,  Acknowledgement Compliant: 3 results) Presentation (13 results) (of which Int'l Joint Research: 2 results,  Invited: 6 results) Funded Workshop (1 results)

  • [Journal Article] Irrational open surfaces of non-negative logarithmic Kodaira dimension2017

    • Author(s)
      Hideo Kojima
    • Journal Title

      Advanced Studies in Pure Mathematics

      Volume: 75

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Notes on the kernels of locally finite higher derivations in polynomial rings2016

    • Author(s)
      Hideo Kojima
    • Journal Title

      Communications in Algebra

      Volume: 44 Issue: 5 Pages: 1924-1930

    • DOI

      10.1080/00927872.2015.1027387

    • Related Report
      2016 Annual Research Report 2014 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Closed polynomials in polynomial rings over integral domains2015

    • Author(s)
      Hideo Kojima and Takanori Nagamine
    • Journal Title

      Journal of Pure and Applied Algebra

      Volume: 219 Issue: 12 Pages: 5493-5499

    • DOI

      10.1016/j.jpaa.2015.05.028

    • Related Report
      2015 Research-status Report
    • Peer Reviewed
  • [Journal Article] Normal log canonical del Pezzo surfaces of rank one with unique singular points2014

    • Author(s)
      Hideo Kojima
    • Journal Title

      Nihonkai Mathematical Journal

      Volume: 25 Pages: 105-118

    • Related Report
      2014 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Presentation] Normal del Pezzo surfaces of rank one with log canonical singular points2017

    • Author(s)
      Hideo Kojima
    • Organizer
      The 15th Affine Algebraic Geometry Meeting
    • Place of Presentation
      関西学院大学大阪梅田キャンパス (大阪府大阪市北区)
    • Year and Date
      2017-03-02
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Some results on open algebraic surfaces of log Kodaira dimension zero2017

    • Author(s)
      小島秀雄
    • Organizer
      第4回代数幾何学研究集会―宇部―
    • Place of Presentation
      宇部工業高等専門学校 (山口県宇部市)
    • Year and Date
      2017-01-07
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Kernels of some higher derivations in polynomial rings2016

    • Author(s)
      小島秀雄
    • Organizer
      第90回米沢数学セミナー「可換Banach環と関連分野研究集会」
    • Place of Presentation
      山形大学工学部 (山形県米沢市)
    • Year and Date
      2016-07-03
    • Related Report
      2016 Annual Research Report
  • [Presentation] Some results on open algebraic surfaces of log Kodaira dimension zero2016

    • Author(s)
      小島秀雄
    • Organizer
      Workshop on Galois point and related topic
    • Place of Presentation
      新潟大学駅南キャンパス ときめいと (新潟県新潟市中央区)
    • Year and Date
      2016-06-04
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Kernels of some higher derivations in polynomial rings2016

    • Author(s)
      小島秀雄
    • Organizer
      第7回多項式環論セミナー
    • Place of Presentation
      首都大学東京
    • Year and Date
      2016-02-18
    • Related Report
      2015 Research-status Report
  • [Presentation] Some results on open algebraic surfaces of non-negative logarithmic Kodaira dimension2015

    • Author(s)
      Hideo Kojima
    • Organizer
      KIAS-TIFR-ICTS Joint Advanced School of Algebraic Surfaces and Related Topics
    • Place of Presentation
      International Centre for Theoretical Sciences of the Tata Institute of Fundamental Research, Bangalore (インド)
    • Year and Date
      2015-11-30
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 3変数多項式環における局所有限高階導分の核2015

    • Author(s)
      小島秀雄
    • Organizer
      射影多様体の幾何とその周辺2015
    • Place of Presentation
      高知大学理学部
    • Year and Date
      2015-10-31
    • Related Report
      2015 Research-status Report
  • [Presentation] オイラー数が0以下となる正規アフィン代数曲面について2015

    • Author(s)
      小島秀雄
    • Organizer
      第6回多項式環論セミナー
    • Place of Presentation
      静岡大学教育学部
    • Year and Date
      2015-08-20
    • Related Report
      2015 Research-status Report
  • [Presentation] 多項式環における局所有限高階導分の核について2015

    • Author(s)
      小島秀雄
    • Organizer
      代数小研究集会 in 新潟
    • Place of Presentation
      新潟大学工学部
    • Year and Date
      2015-06-07
    • Related Report
      2015 Research-status Report
  • [Presentation] 多項式環における局所有限高階導分の核について2015

    • Author(s)
      小島秀雄
    • Organizer
      第5回多項式環論セミナー
    • Place of Presentation
      首都大学東京
    • Year and Date
      2015-01-12
    • Related Report
      2014 Research-status Report
  • [Presentation] Some results on normal log canonical del Pezzo surfaces of rank one2015

    • Author(s)
      小島秀雄
    • Organizer
      代数幾何シンポジウム2014 in 岐阜
    • Place of Presentation
      ソフトピアジャパン センタービル (大垣駅前)
    • Year and Date
      2015-01-10
    • Related Report
      2014 Research-status Report
  • [Presentation] Locally finite iterative higher derivations on k[x,y]2014

    • Author(s)
      小島秀雄
    • Organizer
      アフィン代数幾何学研究集会
    • Place of Presentation
      首都大学東京
    • Year and Date
      2014-09-14
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Open algebraic surfaces of non-negative logarithmic Kodaira dimension2014

    • Author(s)
      Hideo Kojima
    • Organizer
      RIMS Workshop ``Kyoto Workshop on Algebraic Varieties and Automorphism Groups”
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2014-07-11
    • Related Report
      2014 Research-status Report
    • Invited
  • [Funded Workshop] The 15th Affine Algebraic Geometry Meeting2017

    • Place of Presentation
      関西学院大学大阪梅田キャンパス (大阪府大阪市北区)
    • Year and Date
      2017-03-02
    • Related Report
      2016 Annual Research Report

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Published: 2014-04-04   Modified: 2018-03-22  

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