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 $n \geq 2d-4$: 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 $d \leq \frac{e+n}{e+1}$.

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