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Lower bounds for genera of fiber products

Fedor Pakovich

TL;DR

The paper investigates genus lower bounds for components of fiber products of holomorphic maps between compact Riemann surfaces, extending prior results for curves of the form $A(x)-B(y)=0$. Its approach centers on normalization, ramification data, and the Hurwitz automorphism theorem, complemented by a lifting lemma to move to higher-fold fiber products and apply Riemann–Hurwitz. The main results give explicit genus lower bounds for single-component and multi-component fiber products, namely $g( rak E)\ge (g( rak R)-1)( ext{deg }W-1)+1+ rac{ ext{deg }P}{84}$ and $g( rak E)\nge (g( rak R)-1)( ext{deg }V-1)+1+ rac{ ext{deg }P}{ ext{deg }W( ext{deg }W-1) ots ( ext{deg }W- ext{deg }V+1)}$, with sharper statements under tameness and equality cases linked to Hurwitz surfaces. These bounds unify and generalize lower-genus results from rational curves to a broader holomorphic setting, with implications for functional decompositions, Diophantine geometry, and arithmetic dynamics. The paper also proves the sharpness of the main bound via explicit Hurwitz constructions and discusses consequences for irreducibility versus reducibility of fiber-product components.

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.

Lower bounds for genera of fiber products

TL;DR

The paper investigates genus lower bounds for components of fiber products of holomorphic maps between compact Riemann surfaces, extending prior results for curves of the form . Its approach centers on normalization, ramification data, and the Hurwitz automorphism theorem, complemented by a lifting lemma to move to higher-fold fiber products and apply Riemann–Hurwitz. The main results give explicit genus lower bounds for single-component and multi-component fiber products, namely and , with sharper statements under tameness and equality cases linked to Hurwitz surfaces. These bounds unify and generalize lower-genus results from rational curves to a broader holomorphic setting, with implications for functional decompositions, Diophantine geometry, and arithmetic dynamics. The paper also proves the sharpness of the main bound via explicit Hurwitz constructions and discusses consequences for irreducibility versus reducibility of fiber-product components.

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 where and are rational functions.
Paper Structure (10 sections, 9 theorems, 103 equations)

This paper contains 10 sections, 9 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Fiber products and normalizations
  3. Fiber products
  4. Normalizations
  5. Proof of the main results
  6. Upper and lower bounds for $\chi(\EuScript N_V$)
  7. Lifting lemma
  8. Bounds for fiber products with one component
  9. Bounds for fiber products with several components
  10. On the sharpness of the bound of Theorem \ref{['t1']}

Key Result

Theorem 1.1

Let $P:\EuScript R\rightarrow \EuScript C$ and $W:\EuScript T\rightarrow \EuScript C$ be holomorphic maps between compact Riemann surfaces such that the fiber product $(\EuScript R,P)\times_{\EuScript C} (\EuScript T,W)$ consists of a unique component $\EuScript E$ and $g(\EuScript N_W)> 1$. Then

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Lemma 3.4
  • Remark 3.5
  • ...and 1 more