Rational lines on cubic hypersurfaces II

We show that any rational cubic hypersurface of dimension at least 33 defined over a number field $K$ vanishes on a $K$-rational projective line, reducing the previous lower bound of Wooley by two. For $K=\mathbb{Q}$ we can reduce the bound to 29. The main ingredients are a result on linear spaces on quadratic forms over suitable non-real quadratic field extensions, and recent work of Bernert and Hochfilzer on cubic forms over imaginary quadratic number fields for the rational case.