On the number of Mordell–Weil generators for cubic surfaces

Let S be a smooth cubic surface over a field K. It is well-known that new K-rational points may be obtained from old ones by secant and tangent constructions. In this paper we prove, for a cubic surface containing a pair of skew rational lines over a field with at least 13 elements, that the rational points are generated by just one point. We also prove a cubic surface analogue of the unboundedness of ranks conjecture for elliptic curves over the rationals.