  This paper is devoted to a very classical problem that can be summarized as
follows: let S be a non singular compact complex surface, f:S --> P^2 a finite
morphism having simple branching, B the branch curve: to what extent does B
determine f? The problem was first studied by Chisini who proved that B
determines S and f, assuming B to have only nodes and cusps as singularities,
the degree d of f to be greater than 5, and a very strong hypothesis on the
possible degenerations of B, and posed the question if the first or the third
hypothesis could be weakened. Recently Kulikov and Nemirovski proved the result
for d >= 12, and B having only nodes and cusps as singularities. In this paper
we weaken the hypothesis about the singularities of B: we generalize the
theorem of Kulikov and Nemirovski for B having only singularities of type
{x^n=y^m}, in the additional hypothesis of smoothness for the ramification
divisor. Moreover we exhibit a family of counterexamples showing that our
additional hypothesis is necessary.
