The variety of all smooth hypersurfaces of given degree and dimension has the
Fermat hypersurface as a natural base point. In order to study the period map
for such varieties, we first determine the integral polarized Hodge structure
of the primitive cohomology of a Fermat hypersurface (as a module over the
automorphism group of the hypersurface). We then focus on the degree 3 case and
show that the period map for cubic fourfolds as analyzed by R. Laza and the
author gives complete information about the period map for cubic hypersurfaces
of lower dimension dimension.