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 \(d \alpha m\) 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.