On rational cuspidal curves

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.