Mathematische Annalen

, Volume 324, Issue 4, pp 657–673 | Cite as

On rational cuspidal curves

I. Sharp estimate for degree via multiplicities
  • S.Yu. Orevkov
Original article


Let \(C\subset\bold{P}^2\) be a rational curve of degree d which has only one analytic branch at each point. Denote by m the maximal multiplicity of singularities of C. It is proven in [MS] that \(d<3m\). We show that \(d<\alpha m+\const\) where \(\alpha=2.61...\) is the square of the “golden section”. We also construct examples which show that this estimate is asymptotically sharp. When \(\bar{\kappa}(\bold{P}^2-C)=-\infty\), we show that \(d>\alpha m\) and this estimate is sharp. The main tool used here, is the logarithmic version of the Bogomolov-Miyaoka-Yau inequality. For curves as above we give an interpretation of this inequality in terms of the number of parameters describing curves of a given degree and the number of conditions imposed by singularity types.


Rational Curve Main Tool Singularity Type Golden Section Maximal Multiplicity 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • S.Yu. Orevkov
    • 1
  1. 1.Laboratoire E. Picard, UFR MIG, Université Paul Sabatier, 117, route de Narbonne, Toulouse 31062, France FR

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