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

2023 Fiscal Year Final Research Report

• Project/Area Number: 18K03263
• 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: Okayama 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
• Keywords: Zariski pair / plane algebraic curves / curve arrangements / embedded topology

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.

Free Research Field

代数幾何学

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.