On the minimal free resolution of curves of maximal regularity

Let C ⊂ ℙ^r be a nondegenerate projective curve of degree d > r+1 and of maximal regularity. Such curves are always contained in the threefold scroll S(0, 0, r − 2). Also some of such curves are even contained in a rational normal surface scroll. In this paper we study the minimal free resolution of the homogeneous coordinate ring of C in the case where d ≤ 2r − 2 and C is contained in a rational normal surface scroll. Our main result provides all the graded Betti numbers of C explicitly.