The degree of rational cuspidal curves
type: Item
instance of: scholarly article
title: The degree of rational cuspidal curves
zbMATH Open document ID: 0661.14023
DOI: 10.1007/BF01443516
author: Takashi Matsuoka
author: Fumio Sakai
published in: Mathematische Annalen
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164597
review text: Let C be an irreducible complex plane curve. By a \textit{cusp} we mean not only a simple cusp \(y^ 2=x^ 3\) but also any unibranched singular point. We say that C is cuspidal if C has only cusps as its singular points. It turns out that cuspidal curves are rather special. In this article we consider the rational case. For examples of rational cuspidal curves, we refer to \textit{S. Abhyankar} and \textit{T. Moh} [J. Reine Angew. Math. 276, 148-166 (1975; Zbl 0332.14004)], \textit{H. Yoshihara} [Proc. Jap. Acad., Ser. A 55, 152-155 (1979; Zbl 0432.14019)] and \textit{H. Kashiwara} [``Fonctions rationelles de type (0,1) sur l'espace projectif complexe a deux dimensions'', Osaka J. Math. 24, 521-577 (1987)]. We prove the following theorem: if C is a rational cuspidal plane curve of degree \( d,\) then \(d<3\nu\), where \(\nu\) is the maximum of the multiplicities of all cusps. This gives an affirmative answer to the ten years old conjecture of Yoshihara and Tsunoda. Previously, Tsunoda proved the weaker inequality \(d\leq 3\nu +2\) and Yoshihara has settled the cases \(\nu =2\) and \(\nu =3\), \(d\equiv 0\quad mod\quad 3.\) In our proof we use \((i)\quad \det ailed\) analysis of local invariants of cusps, \((ii)\quad \log -Miyaoka\) inequality, \((iii)\quad Zariski's\) theorem on multiple planes.
Mathematics Subject Classification ID: 14H20
Mathematics Subject Classification ID: 13H15
zbMATH DE Number: 4079604
zbMATH Keywords: rational cuspidal curves
zbMATH Keywords: multiplicities
zbMATH Keywords: conjecture of Yoshihara and Tsunoda
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4079604