Let Fq be a finite field with q=8 or q at least 16. Let S be a smooth cubic surface defined over Fq containing at least one rational line. We use a pigeonhole principle to prove that all the rational points on S are generated via tangent and secant operations from a single point.
Let S be a smooth cubic surface defined over a field K. As observed by Segre and Manin, there is a secant and tangent process on S that generates new K-rational points from old. It is natural to ask for the size of a minimal generating set for S(K).
In a recent paper, for fields K with at least 13 elements, Siksek showed that if S contains a skew pair of K-lines then S(K) can be generated from one point. In this paper we prove the corresponding version of this result for fields K having at least 4 elements, and slightly milder results for #K=2 or 3.
