  An old question of Mori asks whether in dimension at least three, any smooth
specialization of a hypersurface of prime degree is again a hypersurface. A
positive answer to this question is only known in degrees two and three. In
this paper, we settle the case of quintic hypersurfaces (in arbitrary
dimension) as well as the case of septics in dimension three. Our results
follow from numerical characterizations of the corresponding hypersurfaces. In
the case of quintics, this extends famous work of Horikawa who analysed
deformations of quintic surfaces.
