On the image of a curve in a normal surface by a plane projection

Abstract

We consider a finite analytic morphism φ = (f, g) defined from a complex analytic normal surface (Z, z) to C2. We describe the topology of the image by φ of a reduced curve on (Z, z) by means of iterated pencils defined recursively for each branch of the curve from the initial one <f, g>. This result generalizes the one obtained in a previous paper for the case in which (Z, z) is smooth and the curve irreducible. The methods we use also permit us to describe the topological type of the discriminant curve of φ, in particular, the topological type of each branch of the discriminant can be obtained from the map without previous knowledge of the critical locus.