Non-uniruledness 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 $P^n$ are not uniruled if $(n+1)/2 \leq d \leq n-3$\mathrm{.}Wealsoshowthatforanypositiveinteger$e$, the space of smooth rational curves of degree $e$ in a general hypersurface of degree $d$ in $P^n$ is not uniruled when $d\ge e\sqrt{n}$.