Abstract
Let X be an arbitrary smooth hypersurface in CPn of degree d. We prove the de Jong–Debarre conjecture for n≥2d−4: the space of lines in X has dimension 2n−d−3. We also prove an analogous result for k-planes: if n≥2d+k−1k+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≥2d! is unirational, and we prove that the space of degree-e curves on X will be irreducible of the expected dimension provided that d≤e+n e+1.
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