Chisini's conjecture for curves with singularities of type $x^n=y^m$

Abstract

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.