Abstract. We compute the Clifford index of all curves on K3 surfaces with Picard group isomorphic to U (m).

Abstract. We study the pencils of minimal degree on the smooth curves lying on a K3 surface X which carries a fixed-point free involution. Generically, the gonality of these curves is totally governed by the genus 1 fibrations of X.

We study a number of natural linear systems carried by any polarized Nikulin surface of genus g, determine their positivity and establish their Brill-Noether theory. As an application, we compute the classes of some natural effective divisors on the moduli space of Nikulin surfaces, relying upon recent work of Farkas and Rimányi.L m = L ⊗ e −m , m = 0, 1, 2, . . .