Linear subspaces of hypersurfaces

Let $X$ be an arbitrary smooth hypersurface in $\mathbb{C}{\mathbb{P}}^n$ of degree $d$. We prove the de Jong-Debarre Conjecture for $n\ge 2d-4$: the space of lines in $X$ has dimension $2n-d-3$. We also prove an analogous result for $k$-planes: if $n\ge 2\left(\genfrac{}{}{0px}{}{d+k-1}{k}\right)+k$, 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 $n \geq 2^{d!}$isunirational,andweprovethatthespaceofdegree$e$curveson$X$willbeirreducibleoftheexpecteddimensionprovidedthat$d \leq \frac{e+n}{e+1}$.