We answer a question of T. Shioda and show that, for any positive integer m prime to 6, the Picard group of the Fermat surface _m is generated by the classes of lines contained in _m. More generally, for any integer m, the classes of lines span a primitive subgroup of Pic_m. These results admit an even further generalization (although not quite straightforward) to Delsarte surfaces. In spite of the algebro-geometric setting, the proof is purely topologi- cal: it revolves about the topology of abelian coverings and the concept of Alexander module of a plane curve.