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We prove that the sweeping components of the space of smooth rational curvesin a smooth hypersurface of degree $d$ in $P^n$ are not uniruled if $(n+1)/2\leq d \leq n-3$. We also show that for any positive integer $e$, the space ofsmooth rational curves of degree $e$ in a general hypersurface of degree $d$ in$P^n$ is not uniruled when $d \geq e \sqrt{n}$.

Non-uniruledness results for spaces of rational curves in hypersurfaces