Classification theory of projective varieties by Galois points and new developments

• Project/Area Number: 16K05088
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Yamagata University
• Principal Investigator: Fukasawa Satoru 山形大学, 理学部, 准教授 (20569496)
• Project Period (FY): 2016-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000); Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: ガロア点 / 準ガロア点 / ガロア群 / 射影 / 射影代数多様体 / 正標数 / ガロワ点 / 自己同型群 / 代数学

Outline of Final Research Achievements

(1) A criterion for the existence of a birational embedding into a projective plane with two Galois points was presented. (2) Several new examples were described, by the criterion (joint works with K. Waki and K. Higashine). (3) The arrangement of Galois lines for the Giulietti-Korchmaros curve was determined (j.w.w. Higahine). (4) A criterion for two Galois points for quotient curves was presented (j.w.w. Higashine). (5) Plane curves possessing two Galois points at which Galois groups generate a semi-direct prodocut were classified (j.w.w. P. Speziali). (6) The set of all Galois points for double-Frobeninus nonclassical curves was determined (j.w.w. H. Borges).

Academic Significance and Societal Importance of the Research Achievements

ガロア点は曲線の対称性を表現していると考えられます。それが複数存在するという状況を、代数関数体という代数学の標準的な言語で表現できること(判定法)を発見しました。その表現は代数幾何、群論、数論という代数学の理論を結びつけるものです。この判定法を、符号(例：ＱＲコード)の構成に用いられている「最大曲線」にも適用し、ガロア点を複数もつことを証明しました。ここにガロア点と符号理論とのつながりが見えます。

Research Products

• [Int'l Joint Research] サンパウロ大学(ブラジル)
• [Int'l Joint Research] バジリカータ大学(イタリア)
• [Journal Article] Birational embeddings of the Hermitian, Suzuki and Ree curves with two Galois points (2019)
  • Author(s): Satoru Fukasawa
  • Journal Title: Finite Fields and Their Applications Volume: 57 Pages: 60-67
  • DOI: 10.1016/j.ffa.2019.02.002
  • Peer Reviewed
• [Journal Article] Galois lines for the Giulietti-Korchmaros curve (2019)
  • Author(s): Satoru Fukasawa and Kazuki Higashine
  • Journal Title: Finite Fields and Their Applications Volume: 57 Pages: 268-275
  • DOI: 10.1016/j.ffa.2019.02.009
  • Peer Reviewed
• [Journal Article] A birational embedding with two Galois points for certain Artin-Schreier curves (2018)
  • Author(s): Satoru Fukasawa and Kazuki Higashine
  • Journal Title: Finite Fields and Their Applications Volume: 52 Pages: 281-288
  • DOI: 10.1016/j.ffa.2018.04.009
  • Peer Reviewed
• [Journal Article] A birational embedding of an algebraic curve into a projective plane with two Galois points (2018)
  • Author(s): Satoru Fukasawa
  • Journal Title: Journal of Algebra Volume: 511 Pages: 95-101
  • DOI: 10.1016/j.jalgebra.2018.06.020
  • Peer Reviewed
• [Presentation] Galois lines for the Giulietti-Korchmaros curve (2018)
  • Author(s): Satoru Fukasawa
  • Organizer: Combinatorics 2018
  • Int'l Joint Research
• [Presentation] Galois lines for the Giulietti-Korchmaros curve (2018)
  • Author(s): 深澤 知
  • Organizer: Workshop on Galois point and related topics
  • Invited
• [Presentation] ２つの外ガロア点に付随する群が半直積を生成する平面曲線の分類 (2018)
  • Author(s): 深澤 知
  • Organizer: 射影多様体の幾何とその周辺 2018
  • Invited
• [Presentation] Galois points for double-Frobenius nonclassical curves (2018)
  • Author(s): 深澤 知
  • Organizer: 新潟代数セミナー
  • Invited
• [Presentation] ２つの外ガロア点に付随する群が半直積を生成する平面曲線の分類 (2018)
  • Author(s): 深澤 知
  • Organizer: 代数幾何ミニワークショップ「正標数における代数曲線」
  • Invited
• [Presentation] 準ガロア点と自己同型群 (2017)
  • Author(s): 深澤 知
  • Organizer: 代数曲線と自己同型群
  • Place of Presentation: 山形大学理学部
  • Year and Date: 2017-03-03
  • Invited
• [Presentation] 平面代数曲線上のガロア点, 有理点, 符号 (2017)
  • Author(s): 深澤 知
  • Organizer: 代数幾何学と暗号数理の展開
  • Place of Presentation: 九州大学西新プラザ
  • Year and Date: 2017-02-07
  • Invited
• [Presentation] 正標数におけるガロア点を複数もつ平面曲線 (2017)
  • Author(s): 深澤 知
  • Organizer: 代数幾何学研究集会―宇部―
  • Place of Presentation: 宇部工業高等専門学校
  • Year and Date: 2017-01-07
  • Invited
• [Presentation] A birational embedding of an algebraic curve into a projective plane with two Galois points (2017)
  • Author(s): Satoru Fukasawa
  • Organizer: The 13th International Conference on Finite Fields and their Applications
  • Int'l Joint Research
• [Presentation] A birational embedding of an algebraic curve into a projective plane with two Galois points (2017)
  • Author(s): 深澤 知
  • Organizer: Workshop on Galois point and related topics
  • Invited
• [Presentation] 正標数におけるガロア点を複数もつ平面曲線 (2017)
  • Author(s): 深澤 知
  • Organizer: Ｋ３曲面・エンリケス曲面ワークショップ
  • Invited
• [Presentation] A birational embedding of an algebraic curve into a projective plane with two Galois points (2017)
  • Author(s): Satoru Fukasawa
  • Organizer: Geometric Galois Theory and Monodromy
  • Int'l Joint Research / Invited
• [Presentation] 標数零における平面曲線のガロア点の個数について (2016)
  • Author(s): 深澤 知
  • Organizer: Workshop on Galois point and related topics
  • Place of Presentation: 新潟大学駅南キャンパスときめいと
  • Year and Date: 2016-06-03
  • Invited
• [Presentation] 有限体上の平面曲線に対する有理点とガロア点 (2016)
  • Author(s): 深澤 知
  • Organizer: 離散数理セミナー
  • Place of Presentation: 山形大学理学部
  • Year and Date: 2016-05-13
  • Invited