Classification theory of projective varieties by Galois points and new developments

2018 Fiscal Year Final Research Report

• 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
• 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

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).

Free Research Field

代数幾何

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。