Studies on the internal structure of algebraic function fields - from the viewpoint of Galois point studies

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K03441
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Niigata University
• Principal Investigator: Takahashi Takeshi 新潟大学, 自然科学系, 准教授 (60390431)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: ガロア点 / 準ガロア点 / ガロア直線 / 同時ガロア点 / 代数関数体の内部構造 / 射影代数多様体の自己同型群

Outline of Final Research Achievements

The study related to "Galois points for algebraic plane curves", introduced by Hisao Yoshihara (Niigata Univ.) have been investigated steadily, with several generalizations.
S. Fukazawa (Yamagata Univ.), K. Miura (Ube National College of Technology) and I defined "quasi-Galois points" for plane curves and studied the groups associated with quasi-Galois points, the number and distribution of quasi-Galois points. J. Komeda (Kanagawa Institute of Technology), S. Kato (Niigata Univ.) and I studied the number and distribution of "Galois lines" for nonsingular curves of degree 6 in 3-dimensional projective space, and determined the maximum numbers and distribution of such lines. A. Ikeda (Niigata Univ.) and I defined "simultaneous Galois points" for reducible and reduced algebraic plane curves. The number of simultaneous Galois points could be determined when each irreducible component of an algebraic curve is a non-singular plane curve not an elliptic curve.

Free Research Field

代数幾何

Academic Significance and Societal Importance of the Research Achievements

射影平面内の代数曲線についての準ガロア点の個数と分布について多くの成果をあげることができ、結果を整理した論文を投稿することができた。３次元射影空間内の射影代数曲線に対するガロア直線について、その本数や分布を調べる方法を開発した。また、既約平面曲線に限らず、可約な平面曲線に対して、そのガロア点である同時ガロア点を定義し、それを研究する基礎を与えた。つまり、本研究課題において、ガロア点に関係する新しい概念をいくつか提供し、それらの基本定理を示すことができた。