  We show that any smooth projective cubic hypersurface of dimension at least
$29$ over the rationals contains a rational line. A variation of our methods
provides a similar result over p-adic fields. In both cases, we improve on
previous results due to the second author and Wooley.
  We include an appendix in which we highlight some slight modifications to a
recent result of Papanikolopoulos and Siksek. It follows that the set of
rational points on smooth projective cubic hypersurfaces of dimension at least
29 is generated via secant and tangent constructions from just a single point.
