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 OP1 (-a -d). OP1 (-b -d) gives the syzigies of the ideal (f0, f1, f2). K[ s, t], and (f0, f1, f2) 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 P (k+1).