ar X iv : a lg - g eo m / 9 40 70 09 v 1 1 5 Ju l 1 99 4 MORDELL – WEIL PROBLEM FOR CUBIC SURFACES

Let V be a plane non–singular geometrically irreducible cubic curve over a finitely generated field k. The Mordell–Weil theorem for V can be restated in the following geometric form: there is a finite subset B ⊂ V (k) such that the whole V (k) can be obtained from B by drawing secants (and tangents) through pairs of previously constructed points and consecutively adding their new intersection points with V. In this note I address the question of validity of this statement for cubic surfaces. After reminding some constructions from the book [Ma], I analyze a numerical example, and then prove a different version of the Mordell–Weil statement for split cubic surfaces. A shameless change of the composition law allows me to reduce this problem to the classical theorem on the structure of abstract projective planes. Unfortunately, the initial question, which is more natural to ask for minimal surfaces, remains unanswered. I would like to call attention to this problem and its calculational aspects. I am grateful to Don Zagier whose tables are quoted in §2, and to M. Rovinsky and A. Skorobogatov, discussions with whom helped me to state and prove the main theorem. §1. A summary of known results 1.1. Notation. Let V be a cubic hypersurface without multiple components over a field k in P d , d ≥ 2. If x, y, z ∈ V (k) are three points (with multiplicities) lying on a line in P d not belonging to V , we write x = y • z. Thus • is a (partial and multivalued) composition law on V (k). We will also consider its restriction on subsets of V (k), e.g. that of smooth points. If x ∈ V (k) is smooth, and does not lie on a hyperplane component of V , the birational map t x : V → V, y → x • y, is well defined. Denote by Bir V the full group of birational automorphisms ov V. The following two results summarize the properties of {t x } for curves and surfaces respectively. The first one is classical, and the second one is proved in [M]. 1.2. Theorem. Let V be a smooth cubic curve. Then: a). Bir V is a semidirect product of the group of projective automorphisms and the subgroup generated by {t x | x ∈ V (k)}. b). We have identically t 2 x = …