  The paper has two parts. First we prove that the specialization maps on
R-equivalence and on the Chow group of zero cycles are isomorphisms for
families over a local, Henselian, Dedekind ring when the special fiber is
smooth and separably rationally connected. This is pretty much the best one can
hope for.
  The second part contains examples of unirational quartic hypersurfaces whith
infinitely many R-equivalence classes. There are also examples of quartics with
points in a large odd degree field extension but no points in any smaller odd
degree extension.
