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

Abstract

We consider a finite analytic morphism $\phi =(f,g)$ defined from a complex analytic normal surface $(Z,z)$ to ${\mathbb{C}}^2$. We describe the topology of the image by $\phi$ 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. As a consequence of the methods we can describe also the topological type of the discriminant curve of $\phi$, in particular the topological type of each branch of the discriminant can be obtained from the map without the previous knowledge of the critical locus.

20 pages, 1 figure