Mordell-Weil Problem for Cubic Surfaces, Numerical Evidence

Abstract

Let V be a plane smooth cubic curve over a finitely generated field k. The Mordell-Weil theorem for V states that there is a finite subset P \subset V(k) such that the whole V(k) can be obtained from P by drawing secants and tangents through pairs of previously constructed points and consecutively adding their new intersection points with V. In this paper we present numerical data regarding the analogous statement for cubic surfaces. For the surfaces examined, we also test Manin's conjecture relating the asymptotics of rational points of bounded height on a Fano variety with the rank of the Picard group of the surface.