On plane algebraic curves having only cusps as their singular points

2014 Fiscal Year Final Research Report

• Project/Area Number: 22540040
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Saitama University
• Principal Investigator: TONO Keita 埼玉大学, 理工学研究科, 非常勤講師 (30422215)
• Co-Investigator(Renkei-kenkyūsha): SAKAI Fumio 埼玉大学, 理工学研究科, 名誉教授 (40036596)
• Project Period (FY): 2010-04-01 – 2015-03-31
• Keywords: 平面曲線 / 特異点

Outline of Final Research Achievements

A plane algebraic curve is called cuspidal if it has only cusps as its singular points, where by a cusp I mean a locally irreducible singular point. For a cuspidal curve C on the complex projective plane let C' denote the proper transform of C via the minimal embedded resolution of the cusps. I obtained some results concerning the self-intersection number of C' for cuspidal plane curves C of genus two having only one cusp. I classified the rational cuspidal plane curves C having two cusps such that the self-intersection number of C' is equal to -1. Then I obtained an upper bound of the self-intersection number of C' for the rational cuspidal plane curves C having at least 3 cusps. Moreover, I determined the rational cuspidal plane curves C having 3 cusps such that the self-intersection number of C' is equal to the bound.

Free Research Field

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