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Hurwitz existence problem and fiber products

Fedor Pakovich

TL;DR

This work advances the Hurwitz existence problem by developing a unified framework based on orbifolds and fiber products of holomorphic maps to systematically explain non-realizability of branch data and to generate new realizability obstructions. Central to the approach are the notions of minimal holomorphic maps between orbifolds, pullback orbifolds, and the fiber product construction, together with Fried’s decomposition results, which yield strong constraints on possible realizations via divisibility and decomposition arguments. The authors derive several broad, uniform non-realizability criteria (Theorems $t_0$–$t_3$), apply them to produce large families of non-realizable data (including connections to planar graphs and dessins d'enfants), and extend the framework to torus-to-sphere maps, thereby illuminating when maps must decompose through orbifold coverings. As a notable consequence, they obtain a Halphen-type theorem for polynomial solutions of $A(z)^a+B(z)^b=C(z)^c$ from the same orbifold perspective. Overall, the paper significantly enhances understanding of when Hurwitz branch data can be realized and clarifies the structure of possible decompositions of rational maps in this context.

Abstract

With each holomorphic map $f: R \rightarrow \mathbb C\mathbb P^1$, where $R$ is a compact Riemann surface, one can associate a combinatorial datum consisting of the genus $g$ of $R$, the degree $n$ of $f$, the number $q$ of branching points of $f$, and the $q$ partitions of $n$ given by the local degrees of $f$ at the preimages of the branching points. These quantities are related by the Riemann-Hurwitz formula, and the Hurwitz existence problem asks whether a combinatorial datum that fits this formula actually corresponds to some map $f$. In this paper, using results and techniques related to fiber products of holomorphic maps between compact Riemann surfaces, we prove a number of results that enable us to uniformly explain the non-realizability of many previously known non-realizable branch data, and to construct a large amount of new such data. We also deduce from our results the theorem of Halphen, proven in 1880, concerning polynomial solutions of the equation $A(z)^a+B(z)^b=C(z)^c$, where $a,b,c$ are integers greater than one.

Hurwitz existence problem and fiber products

TL;DR

This work advances the Hurwitz existence problem by developing a unified framework based on orbifolds and fiber products of holomorphic maps to systematically explain non-realizability of branch data and to generate new realizability obstructions. Central to the approach are the notions of minimal holomorphic maps between orbifolds, pullback orbifolds, and the fiber product construction, together with Fried’s decomposition results, which yield strong constraints on possible realizations via divisibility and decomposition arguments. The authors derive several broad, uniform non-realizability criteria (Theorems –), apply them to produce large families of non-realizable data (including connections to planar graphs and dessins d'enfants), and extend the framework to torus-to-sphere maps, thereby illuminating when maps must decompose through orbifold coverings. As a notable consequence, they obtain a Halphen-type theorem for polynomial solutions of from the same orbifold perspective. Overall, the paper significantly enhances understanding of when Hurwitz branch data can be realized and clarifies the structure of possible decompositions of rational maps in this context.

Abstract

With each holomorphic map , where is a compact Riemann surface, one can associate a combinatorial datum consisting of the genus of , the degree of , the number of branching points of , and the partitions of given by the local degrees of at the preimages of the branching points. These quantities are related by the Riemann-Hurwitz formula, and the Hurwitz existence problem asks whether a combinatorial datum that fits this formula actually corresponds to some map . In this paper, using results and techniques related to fiber products of holomorphic maps between compact Riemann surfaces, we prove a number of results that enable us to uniformly explain the non-realizability of many previously known non-realizable branch data, and to construct a large amount of new such data. We also deduce from our results the theorem of Halphen, proven in 1880, concerning polynomial solutions of the equation , where are integers greater than one.
Paper Structure (10 sections, 26 theorems, 151 equations)

This paper contains 10 sections, 26 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Orbifolds and fiber products
  3. Minimal holomorphic maps between orbifolds
  4. Fiber products
  5. Functional equations and orbifolds
  6. Normalizations and the Fried theorem
  7. Proof of Theorem \ref{['t0']}
  8. Proof of Theorem \ref{['t1']}
  9. Proof of Theorem \ref{['t2']} and Theorem \ref{['t3']}
  10. The Halphen theorem

Key Result

Theorem 1.1

Let $f:{\mathbb C}{\mathbb P}^1 \rightarrow {\mathbb C}{\mathbb P}^1$ be a rational function, and let $\EuScript O$ be a good orbifold such that $f^*(\EuScript O)$ is non-ramified. Then $\chi(\EuScript O) > 0$, and $f = \theta_{\EuScript O} \circ q$ for some rational function $q$. In particular, ${\

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • ...and 17 more