Some results on the local moduli of non-singular normalizations of surfaces with ordinary singularities (Q1107682)

Let S be a compact irreducible analytic surface embedded in a compact threefold W. Assume that S has (with respect to that embedding) at most ordinary singularities, i.e. double points, triple points or cusps. Denote by X the normalization of S, and by \(\Phi_ S\) the sheaf of infinitesimal displacements in W. The authors give a sufficient condition for the connecting homomorphism \[ \delta:\quad H^ 0(S,\Phi_ S)\quad \to \quad H^ 1(X,{\mathcal T}_ X) \] to be surjective. This condition, expressed in terms of the sheaf cohomology of the surfaces S, X and of the ambient threefold W, represents a sharpening of some previous results of the authors [cf. the first author, Sci. Rep. Kagoshima Univ. 32, 23-46 (1983; Zbl 0568.14019)]. Applying this surjectivity criterion in a deformation-theoretical context, the authors compute the Kuranishi families of deformations of complex structures for certain algebraic surfaces of general type, namely for normalizations of surfaces with ordinary singularities in \({\mathbb{P}}^ 3({\mathbb{C}})\). The non- singularity of these Kuranishi families allows to compute the number of moduli for those particular surfaces.