On rigid germs of finite morphisms of smooth surfaces

TL;DR

This paper classifies rigid germs of finite morphisms of smooth surfaces with $ADE$-singularities and links them to Belyi rational functions, revealing a deep connection between surface morphisms and number theory.

Contribution

It establishes a correspondence between rigid germs with $ADE$-singularities and Belyi rational functions, advancing understanding of surface morphism classification.

Findings

Rigid germs correspond to Belyi rational functions.

Germs with $ADE$-singularities are characterized as rigid.

A new link between algebraic geometry and number theory is demonstrated.

Abstract

We prove that a germ of a finite morphism of smooth surfaces is rigid if the germ of its branch curve has one of $ADE$-singularity types and establish a correspondence between the set of rigid germs and the set of Belyi rational functions $f\in \bar{\mathbb Q}(z)$.