Let X be a smooth cubic hypersurface of dimension n≥1 over the
rationals. 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≥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.Comment: 10 pages. Comments very welcom
