Rational Curves on Smooth Cubic Hypersurfaces

We prove that the space of rational curves of a fixed degree on any smooth cubic hypersurface of dimension at least four is irreducible and of the expected dimension. Our methods also show that the space of rational curves of a fixed degree on a general hypersurface in $\mathbb P^n$ of degree 2d ≤ min (n + 4, 2n − 2) and dimension at least three is irreducible and of the expected dimension.