We consider the parametrization (f0, f1, f2) of plane
rational curve c, and we want to relate the splitting type of C (i.e.
the second Betti numbers of the ideal (f0, f1, f2) with the
singularities of the associated Poncelet surface in p3. We are able of doing this for Ascenzi curves, thus generalizing a result in  in the
case of plane curves. Moreover we prove that if the Poncelet surface
s C p3 is singular then it is associated with a curve C which possesses
at least a point of multiplicity >_ 3.