Lower bounds for genera of fiber products

TL;DR

This paper establishes lower bounds for the genera of components in fiber products of holomorphic maps between compact Riemann surfaces, extending classical results on algebraic curves defined by rational functions.

Contribution

It extends existing genus bounds from algebraic curves to fiber products of holomorphic maps between Riemann surfaces, broadening the scope of genus estimation techniques.

Findings

Provides explicit lower bounds for genera of fiber product components

Extends classical algebraic curve genus results to holomorphic maps

Enhances understanding of fiber product topology in complex analysis

Abstract

We give lower bounds for genera of components of fiber products of holomorphic maps between compact Riemann surfaces, extending results on genera of components of algebraic curves of the form $A(x)-B(y)=0,$ where $A$ and $B$ are rational functions.