Linear subspaces of hypersurfaces

Abstract

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 $n\ge 2\left(\genfrac{}{}{0.0pt}{}{d+k-1}{k}\right)+k$, 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 $n\ge 2^{d!}$ is unirational, and we prove that the space of degree-e curves on X will be irreducible of the expected dimension provided that $d\le \frac{e+n}{e+1}$.