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 , whereO P 1( − a − d ) ⊕ O P 1( − 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 ) ( 2 k ≤ 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 .