Rigid germs of finite morphisms of smooth surfaces and rational Belyi pairs

TL;DR

This paper investigates the inverse images of a map from rigid germs of finite morphisms of smooth surfaces to rational Belyi pairs, focusing on monodromies to understand their structure.

Contribution

It introduces a detailed analysis of the inverse images of the map relating rigid germs to Belyi pairs using monodromy representations.

Findings

Characterization of inverse images via monodromy

Connections between singularity types and Belyi pairs

Enhanced understanding of the structure of finite morphisms

Abstract

In \cite{K-rig}, a map $\beta:\mathcal R\to\mathcal{B}el$ from the set $\mathcal R$ of equivalence classes of rigid germs of finite morphisms branched in germs of curves having $ADE$ singularity types onto the set $\mathcal{B}el$ of rational Belyi pairs $f:\mathbb P^1\to\mathbb P^1$ considered up to the action of $PGL(2,\mathbb C)$ was defined. In this article, the inverse images of this map are investigated in terms of monodromies of Belyi pairs.