The degree of rational cuspidal curves. (English) Zbl 0661.14023
Let C be an irreducible complex plane curve. By a 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 S. Abhyankar and T. Moh [J. Reine Angew. Math. 276, 148-166 (1975; Zbl 0332.14004)], H. Yoshihara [Proc. Jap. Acad., Ser. A 55, 152-155 (1979; Zbl 0432.14019)] and 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.
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.
Reviewer: T.Matsuoka
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