Let X be an arbitrary smooth hypersurface in of degree d. We prove the de Jong–Debarre conjecture for : the space of lines in X has dimension . We also prove an analogous result for k-planes: if , 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 is unirational, and we prove that the space of degree-e curves on X will be irreducible of the expected dimension provided that .
"Linear subspaces of hypersurfaces." Duke Math. J. 170 (10) 2263 - 2288, 15 July 2021. https://doi.org/10.1215/00127094-2021-0035