I don't know if there is an elementary proof out there which works for all hypersurfaces of degree $d\ge n$, but if $X$ is general, you can see quite easily that they contain projective lines - or equivalently, the Fano variety of lines is of positive (expected) dimension. This is proved in Joe Harris' algebraic geometry book.

In fact, the following theorem is proved in M. Hochster and D. Laksov, *The Linear Syzygies of Generic Forms*, Comm. Algebra 15 (1987) and the proof is elementary. Here $F_k(X)$ denotes the Fano variety of $k-$planes in $X$ (thus if $F_1(X)$ is non-empty then your $X$ contains lines.

**Theorem**. For $n\ge k\ge 0$ and $d \ge 3$, let $\phi= (k + 1)(n − k)-{k+d
> \choose d}$.

a) if $\phi <0$ the subvariety of
hypersurfaces that contain a $k$-plane
has codimension $−\phi$.

b) For $\phi=0$ every hypersurface of
degree $d$ in $P^n$ contains a
k-plane, and a general hypersurface
contains a positive number of
k-planes.

c) For $\phi>0$, a hypersurface $X$
has $\dim(F_k(X))$ \ge \phi$ with
equality for general X.

See also Joe Harris' algebraic geometry book and Alex Waldron's thesis and the references given there.

coveredby rational curves. I don't know of a proof other than the Frobenius trick, but I'm certainly not up to date on this. – Jack Huizenga Aug 10 '11 at 9:58