  The goal of this paper is to give a method to compute the space of
infinitesimal deformations of a double cover of a smooth algebraic variety. The
space of all infinitesimal deformations has a representation as a direct sum of
two subspaces. One is isomorphic to the space of simultaneous deformations of
the branch locus and the base of the double covering. The second summand is the
subspace of deformations of the double covering which induce trivial
deformations of the branch divisor. The main result of the paper is a
description of the effect of imposing singularities in the branch locus.
  As a special case we study deformations of Calabi--Yau threefolds which are
non--singular models of double cover of the projective 3--space branched along
an octic surface. We show that in that case the number of deformations can be
computed explicitly using computer algebra systems. This gives a method to
compute the Hodge numbers of these Calabi--Yau manifolds. In this case the
transverse deformations are resolutions of deformations of double covers of
projective space but not double covers of a blow--up of projective space. In
the paper we gave many explicit examples.
