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To classify planar complex rational cuspidal curves    E  ⊆   P  2it remains to classify the ones with complement of log general type, that is, the ones for which    κ  (   K  X   +  D  )  =  2   , where    (  X  ,  D  )    is a log resolution of    (   P  2   ,  E  )   . It is conjectured that    κ  (   K  X   +   1  2   D  )  =  −  ∞    and henceP  2   ∖  E    is    C   ∗  ∗    ‐fibered, whereC   ∗  ∗    =   C  1   ∖   {  0  ,  1  }    , or    −  (   K  X   +   1  2   D  )    is ample on some minimal model of    (  X  ,   1  2   D  )   . 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.

Classification of planar rational cuspidal curves I. C ∗ ∗ ‐fibrations