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Commutative algebraic study of hyperplane arrangements

Research Project

Project/Area Number 18F18756
Research Category

Grant-in-Aid for JSPS Fellows

Allocation TypeSingle-year Grants
Section外国
Research Field Algebra
Research InstitutionHokkaido University

Principal Investigator

吉永 正彦  北海道大学, 理学研究院, 教授 (90467647)

Co-Investigator(Kenkyū-buntansha) PALEZZATO ELISA  北海道大学, 理学(系)研究科(研究院), 外国人特別研究員
Project Period (FY) 2018-11-09 – 2021-03-31
Project Status Completed (Fiscal Year 2020)
Budget Amount *help
¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 2020: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2019: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2018: ¥600,000 (Direct Cost: ¥600,000)
Keywords超平面配置 / 自由性
Outline of Annual Research Achievements

超平面配置は様々な組み合わせ論的問題と関係した研究対象であるが、同時に可換環論的な側面も持っている。そのような代数的構造の代表的なものの一つに対数的ベクトル場のなす加群がある。ここ数年、様々な数列の対数的凸性との関連で、Gorenstein-Artin環のLefschetz性(なめらかな射影多様体のコホモトジー環においてKahler形式から定まる元が持つ非退化性(Lefschetz分解)や正値性(Hodge-Riemann不等式)を代数的に抽象化したもの)が注目を集め、活発に研究されている。超平面配置の対数的ベクトル場の加群は、超平面配置のヤコビイデアルやその商環であるヤコビ環の代数的性質と密接に関係しており、日本学術振興会外国人特別研究員のElisa Palezzato氏は、超平面配置の可換環論的な側面について、いくつかの研究を進めている。一つ目のテーマはヤコビ環の可換間論的な側面、とくにLefschetz性についてである。超平面配置の特異点は孤立特異点でないため、ヤコビ環はArtin環とはならない。そこで自然数k>0に対して「k-Lefschetz性」という概念が定義されている。超平面配置のヤコビ間がk-Lefschetz性を持つための必要十分条件をいくつか明らかにした。二つ目のテーマは、数年前に阿部拓郎氏によって導入された "Plus-one generated" という性質をもった超平面配置である。これは寺尾宏明氏を中心に長年よく調べられてきた「自由配置」の次の自然なクラスト考えられている。また、有理数体上定義された超平面配置を mod p で有限体上の超平面配置としたときに自由性がどうふるまうかという基本的な問題にも成果を挙げた。
これら超平面配置の研究のほかに、グラフ理論の研究者とともに、純粋にグラフ理論の研究も進めている。

Research Progress Status

令和2年度が最終年度であるため、記入しない。

Strategy for Future Research Activity

令和2年度が最終年度であるため、記入しない。

Report

(3 results)
  • 2020 Annual Research Report
  • 2019 Annual Research Report
  • 2018 Annual Research Report
  • Research Products

    (15 results)

All 2020 2019 2018 Other

All Int'l Joint Research (1 results) Journal Article (7 results) (of which Int'l Joint Research: 7 results,  Peer Reviewed: 7 results,  Open Access: 2 results) Presentation (7 results) (of which Int'l Joint Research: 4 results,  Invited: 7 results)

  • [Int'l Joint Research] ジェノア大学(イタリア)

    • Related Report
      2018 Annual Research Report
  • [Journal Article] Lefschetz properties and hyperplane arrangements2020

    • Author(s)
      Palezzato Elisa、Torielli Michele
    • Journal Title

      Journal of Algebra

      Volume: 555 Pages: 289-304

    • DOI

      10.1016/j.jalgebra.2020.02.039

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Localization of plus-one generated arrangements2020

    • Author(s)
      Palezzato Elisa、Torielli Michele
    • Journal Title

      Communications in Algebra

      Volume: 49 Issue: 1 Pages: 301-309

    • DOI

      10.1080/00927872.2020.1798976

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Free hyperplane arrangements over arbitrary fields2019

    • Author(s)
      Palezzato Elisa、Torielli Michele
    • Journal Title

      Journal of Algebraic Combinatorics

      Volume: 0 Issue: 2 Pages: 1-2

    • DOI

      10.1007/s10801-019-00901-x

    • Related Report
      2020 Annual Research Report 2019 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] 3-tuple total domination number of rook's graphs2019

    • Author(s)
      Pahlavsay Behnaz、Palezzato Elisa、Torielli Michele
    • Journal Title

      Discussiones Mathematicae Graph Theory

      Volume: 0 Pages: 1-2

    • DOI

      10.7151/dmgt.2242

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Computing and Using Minimal Polynomials2019

    • Author(s)
      Abbott, Bigatti, Palezzato, Robbiano
    • Journal Title

      Journal of Symbolic Computation

      Volume: 印刷中

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] New characterizations of freeness for hyperplane arrangements2019

    • Author(s)
      Bigatti, Palezzato, Torielli
    • Journal Title

      Journal of Algebraic Combinatorics

      Volume: 印刷中 Issue: 2 Pages: 297-315

    • DOI

      10.1007/s10801-019-00876-9

    • NAID

      120006980042

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Hyperplane arrangements in CoCoA2019

    • Author(s)
      Palezzato, Torielli
    • Journal Title

      Journal of Software for Algebra and Geometry

      Volume: 印刷中

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Presentation] Modular methods for hyperplane arrangements2019

    • Author(s)
      Elisa Palezzato
    • Organizer
      ACA 2019 Algebraic Geometry from an algorithmic point of view
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Modular approach on hyperplane arrangements2019

    • Author(s)
      Elisa Palezzato
    • Organizer
      Hyperplane arrangements in Wakkanai
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Lefschetz properties and hyperplane arrangements2019

    • Author(s)
      Elisa Palezzato
    • Organizer
      Discrete Geometric Structure Seminar
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] On the Freeness of Hyperplane Arrangements over Arbitrary Fields2019

    • Author(s)
      Elisa Palezzato
    • Organizer
      Hyperplane arrangements, configuration spaces and related topics
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On the Freeness of Hyperplane Arrangements over Arbitrary Fields2019

    • Author(s)
      Elisa Palezzato
    • Organizer
      Branched Coverings, Degenerations, and Related Topics 2019
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Free hyperplane arrangements over arbitrary fields and their computation with CoCoA2018

    • Author(s)
      Elisa Palezzato
    • Organizer
      信州大学トポロジーセミナー
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Free hyperplane arrangements over arbitrary fields and their computation with CoCoA2018

    • Author(s)
      Elisa Palezzato
    • Organizer
      RIMS Conference "Computer Algebra; Theory and its Applications"
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited

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Published: 2018-11-12   Modified: 2024-03-26  

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