Classification of planar rational cuspidal curves. I. C**-fibrations

To classify complex rational cuspidal curves $E\subseteq {\mathbb{P}}^2$ it remains to classify the ones with complement of log general type, i.e. the ones for which $\kappa (K_X+D)=2$, where $(X,D)$ is a log resolution of $({\mathbb{P}}^2,E)$. It is conjectured that $\kappa (K_X+\frac{1}{2}D)=-\mathrm{\infty }$ and hence ${\mathbb{P}}^2\setminus E$ is ${\mathbb{C}}^{\ast \ast }$-fibered, where ${\mathbb{C}}^{\ast \ast }={\mathbb{C}}^1\setminus \{0,1\}$, or $-(K_X+\frac{1}{2}D)$ is ample on some minimal model of $(X,\frac{1}{2}D)$. Here we classify, up to a projective equivalence, those rational cuspidal curves for which the complement is ${\mathbb{C}}^{\ast \ast }$-fibered. From the rich list of known examples only very few are not of this type. We also discover a new infinite family of bicuspidal curves with unusual properties.