Lines on hypersurfaces

Given an arbitrary smooth hypersurface X of degree d in P^n, a natural object of study is the space of lines on X. The de Jong-Debarre Conjecture predicts that for n at least d, the space of lines has dimension 2n-d-3 for any such X. We prove this conjecture for n > 2d-5. In fact, in the case n > 2d-2, we prove that the space of lines is irreducible. Time permitting, we discuss a generalization of the result to k-planes in X, applications to the spaces of rational curves in X of low degree, and the study of the unirationality of X. This is joint work with Roya Beheshti.