by   S.Yu. Orevkov

Abstract:  Let  C  be a rational curve of degree  d  in the projective plane $P^2$ which has only one analytic branch at each point. Denote by  m  the maximal multiplicity of singularities of  C . It is proved 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 $k(P^2-C)=-\infty$, we show that $d > \alpha m$ and this estimate is sharp where  k  denotes the logarithmic Kodaira dimension.
      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.

[MT].  T.Matsuoka, F.Sakai, The degree of rational cuspidal curves, Math. Ann.,  285(1989), 233-247