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

TL;DR

This paper analyzes the topology of the image of a reduced curve under a finite analytic morphism from a normal surface to complex plane, generalizing previous results from smooth surfaces and irreducible curves.

Contribution

It extends the understanding of the topological image of curves under finite morphisms to normal surfaces, including the discriminant curve's topological type, without prior knowledge of the critical locus.

Findings

Describes the topology of the image of a reduced curve via iterated pencils for each branch.

Generalizes previous results from smooth, irreducible cases to normal surfaces.

Provides a method to determine the topological type of discriminant curve branches.

Abstract

We consider a finite analytic morphism $\varphi =(f,g)$ defined from a complex analytic normal surface $(Z,z)$ to ${\mathbb C}^2$. We describe the topology of the image by $\varphi$ of a reduced curve on $(Z,z)$ by means of iterated pencils defined recursively for each branch of the curve from the initial one $\langle f,g \rangle$. 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 $\varphi$, 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.