  In this article we give a general approach to the following analogue of
Shafarevich's conjecture for some polarized algebraic varieties; suppose that
we fix a type of an algebraic variety and look at families of such type of
varieties over a fixed Riemann surface with fixed points over which we have
singular varieties, then one can ask if the set of such families, up to
isomorphism, is finite.
  In this paper we give a general approach to such types of problems. The main
observation is the following; suppose that the moduli space of a fixed type of
algebraic polarized variety exists and suppose that in some projective smooth
compactification of the coarse moduli the discriminant divisor supports an
ample one, then it is not difficult to see that this fact implies the analogue
of Shafarevich's conjecture.
  In this article we apply this method to certain polarized algebraic K3
surfaces and also to Enriques surfaces.
