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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

Abstract.

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.

Keywords

Rational Curve Main Tool Singularity Type Golden Section Maximal Multiplicity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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