Generators for cubic surfaces with two skew lines over finite fields (Q2376315)

Let \(S\) be a smooth cubic surface over a field \(K\). There is a secant and tangent process analogous to the one that, when \(K\) is finite, provides a structure of group to the set of \(K\)-rational points of an elliptic curve over \(K\) (this group is either cyclic or isomorphic to a product of two cyclic subgroups). This process was introduced by Segre and studied by several authors, most notably Manin. It does not give the set of \(K\)-rational points of \(S\) a group structure, but it is natural to ask whether it is still possible to generate all the \(K\)-rational points from just one or two points. Recently, \textit{S. Siksek} [J. Number Theory 132, No. 11, 2610--2629 (2012; Zbl 1253.14023)] proved that \(S(K)\) can be generated by just one point under the following assumptions: the field \(K\) has, at least, 13 elements and \(S\) contains a skew pair of lines defined over \(K\). The author of the paper under review, in Theorem 1, extends this result for fields of, at least, 4 elements. In addition, in Theorems 2 and 3, similar (but slightly weaker) statements are proved over fields with 2 or 3 elements. More specifically, in the case of fields with 3 elements one must add the following additional condition: each line of the skew pair of lines contains at most one \(K\)-rational Eckardt point (that is, a point where three of the 27 lines meet); in the case of fields of 2 elements one must suppose that \(S\) contains a line defined over \(K\) that does not contain any \(K\)-rational Eckardt point. The proof of Theorem 1 is theoretical but the proofs of Theorems 2 and 3 use a certain amount of exhaustive computer enumeration.