  Using Voisin's method we prove that a very general hypersurface of degree at
least 4 in complex projective space of dimension 6, 7, 8 or 9 is not stably
rational and so, in particular, not rational. We obtain the same conclusion for
the double covering of projective space of dimension 6, 7, 8 or 9, branched
along a very general quartic hypersurface. On the other hand, such double
coverings as well as general quartic hypersurfaces of dimension at least 5 are
known to be unirational.
