  Let X be a smooth genus g curve equipped with a simple morphism f: X -> C,
where C is either the projective line or more generally any smooth curve whose
gonality is computed by finitely many pencils. Here we apply a method developed
by Aprodu to prove that if g is big enough then X satisfies both Green and
Green-Lazarsfeld conjectures. We also partially address the case in which the
gonality of C is computed by infinitely many pencils.
