  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).
