On parameterizations of plane rational curves and their syzygies

Let C be a plane rational curve of degree d and p : ˜C →C be its normalization.We are interested in the splitting type (a, b) of C, where O_P1 (−a − d) ⊕ O_P1 (−b − d) gives the syzigies of the ideal ( f_0, f_1, f_2) ⊂ K[s, t], and ( f_0, f_1, f_2) is a parameterization of C. We want to describe in which cases (a, b) = (k, d − k) (2k ≤ d), via a geometric description; namely we show that (a, b) = (k, d − k) if and only if C is the projection of a rational curve on a rational normal surface in Pk+1.