  Let S be a smooth, projective surface of Picard rank 1 and very ample
generator embedding S into P^n. Let C be a smooth curve in O(m) for m \geq 5.
We prove that any base-point free, complete g^r_d on C for r\in\{1,2\} and d
small enough is cut out by a hyperplane section restricted to a multisecant
(n-r-1)-plane.
