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New developments in the study of the embedded topology of plane algebraic curves

Research Project

Project/Area Number 18K03263
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 11010:Algebra-related
Research InstitutionOkayama University of Science (2021-2023)
Ibaraki National College of Technology (2018-2020)

Principal Investigator

Bannai Shinzo  岡山理科大学, 理学部, 准教授 (20732556)

Project Period (FY) 2018-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2022: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2021: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2020: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2018: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
KeywordsZariski pair / plane algebraic curves / curve arrangements / embedded topology / 平面曲線の埋め込み位相 / 平面曲線配置 / Zariski tuple / 分解曲線 / 楕円曲線 / 楕円曲面 / 曲線配置 / 平面曲線 / ザリスキー対 / マトロイド / 対数的べクトル場
Outline of Final Research Achievements

In this project, the embedded topology of algebraic plane curves was studied. The main interest was in what kind of algebraic properties of plane curves lead to differences in the embedded topology, and how can we describe the subtle differences in the algebraic properties in a more manageable way. As a result, the relation between the formerly used invariant called "splitting types" and concepts such as two-graphs and the torsion points of Jacobians were found. Also, as an application many new interesting curve arrangements were found. These results were compiled into 10 papers, published in refereed research journals and lead to joint international research with researchers from abroad.

Academic Significance and Societal Importance of the Research Achievements

平面代数曲線の埋め込み位相の研究における究極的な目標は, 完全な分類を与えることであるが, 現時点では目標の到達には程遠いのが現状である. 本研究の成果により, 直接扱うのが難しい曲線の位相的な特徴を代数的な特徴として捉え, さらには代数的な差異を今までより簡明に記述することができる様になった. その結果, これまで位相的な特徴が把握できていなかった曲線についての理解が進み, 次数が低い曲線をある程度扱うことができる様になり, 大目標へ僅かではあるが近づくことが出来た.

Report

(7 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • Research Products

    (30 results)

All 2024 2023 2022 2021 2020 2019 2018 Other

All Int'l Joint Research (5 results) Journal Article (10 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 10 results,  Open Access: 2 results) Presentation (14 results) (of which Int'l Joint Research: 4 results,  Invited: 11 results) Funded Workshop (1 results)

  • [Int'l Joint Research] University of Zaragoza(スペイン)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] Shamoon College of Engineering/Tel Aviv University(イスラエル)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] University of Zaragoza(スペイン)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] Shamoon College of Engineering/Tel Aviv University(イスラエル)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] Universidad de Zaragoza(スペイン)

    • Related Report
      2020 Research-status Report
  • [Journal Article] Ramified and Split Models of Elliptic Surfaces and Bitangent Lines of Quartic Curves2023

    • Author(s)
      S. Bannai, H. Tokunaga and E. Yorisaki
    • Volume
      71
    • Pages
      51-69
    • DOI

      10.14992/0002000760

    • URL

      https://rikkyo.repo.nii.ac.jp/records/2000760

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Torsion divisors of plane curves and Zariski pairs2022

    • Author(s)
      E. Artal Barolo, Sh. Bannai, T. Shirane and H. Tokunaga
    • Journal Title

      Algebra i Analiz

      Volume: 34:5

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Trisections on certain rational elliptic surfaces and families of Zariski pairs degenerating to the same conic-line arrangement2022

    • Author(s)
      Bannai S.、Kawana N.、Masuya R.、Tokunaga H.
    • Journal Title

      Geometriae Dedicata

      Volume: 216 Issue: 1

    • DOI

      10.1007/s10711-021-00672-5

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] Elliptic surfaces of rank one and the topology of cubic-line arrangements2021

    • Author(s)
      S. Bannai and H. Tokunaga
    • Journal Title

      J. Number Theory

      Volume: 221 Pages: 174-189

    • DOI

      10.1016/j.jnt.2020.06.005

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Torsion divisors of plane curves with maximal flexes and Zariski pairs2020

    • Author(s)
      E. Artal Bartolo, S. Bannai, T. Shirane and H. Tokunaga
    • Journal Title

      accepted to Math. Nachr.

      Volume: -

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Two-graphs and the embedded topology of smooth quartics and its bitangent lines2020

    • Author(s)
      BANNAI Shinzo、OHNO Momoko
    • Journal Title

      Canadian Mathematical Bulletin

      Volume: - Issue: 4 Pages: 1-13

    • DOI

      10.4153/s0008439520000053

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Zariski N-ples for a smooth cubic and its tangent lines2020

    • Author(s)
      S. Bannai and H. Tokunaga
    • Journal Title

      Proc. Japan Acad. Ser. A Math. Sci.

      Volume: 96 Issue: 2 Pages: 18-21

    • DOI

      10.3792/pjaa.96.004

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] ThematroidstructureofvectorsoftheMordell-Weillatticeandthetopologyofplane quartics and bitangent lines2019

    • Author(s)
      R. Sato、S. Bannai
    • Journal Title

      Monografias Matematicas Garcia de Galdeano

      Volume: 42 Pages: 265-274

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Rational points of elliptic surfaces and the topology of cubic-line, cubic-conic-line arrangements2019

    • Author(s)
      S. Bannai, H. Tokunaga and M. Yamamoto
    • Journal Title

      Hokkaido Mathematical Journal

      Volume: 印刷中

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] A note on the topology of arrangements for a smooth plane quartic and its bitangent lines2019

    • Author(s)
      S. Bannai, H. Tokunaga and M. Yamamoto
    • Journal Title

      Hiroshima Mathematical Journal

      Volume: 印刷中

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Presentation] The realization space of a certain conic line arrangement of degree 7 and a π_1-equivalent Zariski pair2023

    • Author(s)
      Shinzo Bannai
    • Organizer
      Work Shop on Algebraic Geometry and Topology 2023
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Ramified and Split models of rational elliptic surfaces and bitangent lines for a quartic curve2023

    • Author(s)
      寄崎 恵美子, 坂内 真三, 徳永浩雄
    • Organizer
      日本数学会2023年度年会
    • Related Report
      2022 Research-status Report
  • [Presentation] Splitting invariants and a π_1-equivalent Zariski pair of conic-line arrangements2023

    • Author(s)
      白根 竹人, M. Amram, 坂内 真三, U. Sinichkin, 徳永 浩雄
    • Organizer
      日本数学会2023年度年会
    • Related Report
      2022 Research-status Report
  • [Presentation] Splitting invariants and a π_1-equivalent Zariski-pair of conic-line arrangements2022

    • Author(s)
      坂内 真三
    • Organizer
      城崎代数幾何学シンポジウム
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Ramified and split models of rational elliptic surfaces and the geometry of quartics and bitangent lines2022

    • Author(s)
      坂内 真三
    • Organizer
      第26回代数曲面論ワークショップ
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Splitting invariants and a π_1-equivalent Zariski-pair of conic-line arrangements2022

    • Author(s)
      坂内 真三
    • Organizer
      神戸代数幾何学ワークショップ
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] 射影平面曲線の埋め込み位相の分類問題2021

    • Author(s)
      坂内 真三
    • Organizer
      半田山・幾何・代数セミナー
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] 射影平面曲線の埋め込み位相の分類問題とその展開2020

    • Author(s)
      坂内 真三
    • Organizer
      徳島大学数学談話会
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] Torsion divisors of plane curves and Zariski pairs2020

    • Author(s)
      白根竹人, E.Artal Bartolo, 坂内 真三, 徳永 浩雄
    • Organizer
      日本数学会 2020年度年会
    • Related Report
      2019 Research-status Report
  • [Presentation] The topology of cubic-line arrangements2019

    • Author(s)
      Shinzo Bannai
    • Organizer
      Geometries in Pyrenees
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Torsion divisors of plane curves and Zariski pairs2019

    • Author(s)
      Shinzo Bannai
    • Organizer
      第17回代数曲線論シンポジウム
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] Torsion divisors of plane curves and Zariski pairs2019

    • Author(s)
      Shinzo Bannai
    • Organizer
      湯布院代数幾何学ワークショップ
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] Matroids, two-graphs and the embedded topology of quartics and bitantgent lines2018

    • Author(s)
      S. Bannai
    • Organizer
      Universidad Complutense Madrid, Seminario de Algebra, Geometria y Topologia
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Matroids, two-graphs and the embedded topology of quartics and bitantgent lines2018

    • Author(s)
      S. Bannai
    • Organizer
      Fifteenth International Conference Zaragoza-Pau on Mathematics and its Applications
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Funded Workshop] 第8回 代数幾何学研究集会‐宇部‐2024

    • Related Report
      2023 Annual Research Report

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Published: 2018-04-23   Modified: 2025-01-30  

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