Infinitesimal deformations of double covers of smooth algebraic varieties

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.