On the quasi-group of a cubic surface over a finite field

We study the Mordell–Weil group MW(V) for cubic surfaces V over finite fields that are not necessarily irreducible and smooth. We construct a surjective map from MW(V) to a group that can be computed explicitly. For #MW(V), this yields a lower bound, which is (often but) not always trivial. To distinguish cases, we follow the classification of cubic surfaces, originally due to Schläfli and Cayley. On the other hand, we describe an algorithm that a priori gives an upper bound for MW(V). We report on our experiments for “randomly” chosen surfaces of the various types, showing that in all but one case lower and upper bounds agree. Finally, we give two applications to the number field case. First, we prove that the number of generators of MW(V) is unbounded. A second application explains why, for many reduction types, the Brauer–Manin obstruction may not distinguish points reducing to the smooth part.