Nonuniruledness results for spaces of rational curves in hypersurfaces

We prove that the sweeping components of the space of smooth rational curves in a smooth hypersurface of degree d in ℙⁿ are not uniruled if (n + 1)/2 ≤ d ≤ n - 3. We also show that for any e ≥ 1, the space of smooth rational curves of degree e in a general hypersurface of degree d in ℙⁿ is not uniruled roughly when d ≥ e√n.