Fano varieties and linear sections of hypersurfaces

Under a hypothesis on $k$, $d$ and $n$ that is almost the best possible, we prove that for every smooth degree $d$ hypersurface in $P^n$, the $k$-plane sections dominate the moduli space of degree $d$ hypersurface in $P^k$. Using this we prove rational simple connectedness of every smooth degree $d$ hypersurface in $P^n$, under a suitable hypothesis on $d$ and $n$ (previous results were only for general hypersurfaces).