A Mordell–Weil theorem for cubic hypersurfaces of high dimension

Abstract

Let $X∕\mathbb{Q}$ be a smooth cubic hypersurface of dimension $n\ge 1$. It is well-known that new rational points may be obtained from old ones by secant and tangent constructions. In view of the Mordell–Weil theorem for $n=1$, Manin (1968) asked if there exists a finite set $S$ from which all other rational points can be thus obtained. We give an affirmative answer for $n\ge 48$, showing in fact that we can take the generating set $S$ to consist of just one point. Our proof makes use of a weak approximation theorem due to Skinner, a theorem of Browning, Dietmann and Heath-Brown on the existence of rational points on the intersection of a quadric and cubic in large dimension, and some elementary ideas from differential geometry, algebraic geometry and numerical analysis.