  Let $X$ be an arbitrary smooth hypersurface in $\mathbb{C} \mathbb{P}^n$ of
degree $d$. We prove the de Jong-Debarre Conjecture for $n \geq 2d-4$: the
space of lines in $X$ has dimension $2n-d-3$. We also prove an analogous result
for $k$-planes: if $n \geq 2 \binom{d+k-1}{k} + k$, then the space of
$k$-planes on $X$ will be irreducible of the expected dimension. As
applications, we prove that an arbitrary smooth hypersurface satisfying $n \geq
2^{d!}$ is unirational, and we prove that the space of degree $e$ curves on $X$
will be irreducible of the expected dimension provided that $d \leq
\frac{e+n}{e+1}$.
