Hypersurfaces with too many rational curves

We study smooth hypersurfaces of degree d ≥ n + 1 in P n whose spaces of smooth rational curves of low degrees are larger than expected, and show that under certain conditions, the primitive part of the middle cohomology of such hypersurfaces have non-trivial Hodge substructures. As an application, we prove that the space of lines on any smooth Fano hypersurface of degree d ≤ 8 in P n has the expected dimension 2 n - d - 3 .