$n-2$-spaces on cubic hypersurfaces

Question

Let $X\subset\mathbb{P}^n$ be a smooth cubic hypersurface. Is there, as in the case of $n=3$, an $n-2$-dimensional linear subspace contained in $X$? More general, is there a good description of the Picard group of $X$ for some $n\geq 4$?

Answer 1

By Lefschetz theorem for $n \ge 3$ the Picard group of a smooth hypersurface is generated by the class of a hyperplane section.

Similarly, for $n \ge 5$ (again by Lefschetz theorem) the Chow group of codimension 2 cycles is generated by the class of a codimension 2 linear section; in particular a smooth hypersurface of dimension $n \ge 5$ has no codimension 2 linear subspaces.

In dimension $n = 4$ the moduli space of smooth cubic hypersurfaces (which is 20-dimensional) contains an irreducible divisor of hypersurfaces containing a plane. Thus a general smooth cubic 4-fold has no planes, while some special smooth cubic 4-folds have.