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

Abstract We consider a finite analytic morphism $$\varphi =(f,g)$$ φ = ( f , g ) defined from a complex analytic normal surface ( Z , z ) to $$\mathbb {C}^2$$ C 2 . We describe the topology of the image by $$\varphi $$ 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 $$\langle f,g \rangle $$ ⟨ 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 $$\varphi $$ 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.