# On germs of finite morphisms of smooth surfaces

## TL;DR

This paper classifies four-sheeted finite morphism germs between smooth complex surfaces up to smooth deformation, analyzing their branch curve singularities and local monodromy groups.

## Contribution

It provides a classification of four-sheeted finite morphism germs of smooth surfaces, including their singularities and monodromy, advancing understanding of their deformation behavior.

## Key findings

- Classification of four-sheeted germs up to smooth deformation
- Analysis of branch curve singularities
- Investigation of local monodromy groups

## Abstract

Questions related to deformations of germs of finite morphisms of smooth surfaces are discussed. A classification of the four-sheeted germs of finite covers $F: (U,o')\to (V,o)$ is given up to smooth deformations, where $(U,o')$ and $(V,o)$ are two connected germs of smooth complex-analytic surfaces. The singularity types of their branch curves and the local monodromy groups are investigated also.