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

To classify planar complex rational cuspidal curves EP2 it remains to classify the ones with complement of log general type, that is, the ones for which (KX+D)=2, where (X,D) is a log resolution of (P2,E). It is conjectured that (KX+12D)=- and hence P2\E is C-fibered, where C=C1\{0,1}, or -(KX+12D) is ample on some minimal model of (X,12D). Here we classify, up to a projective equivalence, those rational cuspidal curves for which the complement is C-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.