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Monodromy groups of indecomposable coverings of bounded genus

Danny Neftin, Michael E. Zieve

Abstract

For each nonnegative integer $g$, we classify the ramification types and monodromy groups of indecomposable coverings of complex curves $f: X\to Y$ where $X$ has genus $g$, under the hypothesis that $n:=°(f)$ is sufficiently large and the monodromy group is not $A_n$ or $S_n$. This proves a conjecture of Guralnick and several conjectures of Guralnick and Shareshian.

Monodromy groups of indecomposable coverings of bounded genus

Abstract

For each nonnegative integer , we classify the ramification types and monodromy groups of indecomposable coverings of complex curves where has genus , under the hypothesis that is sufficiently large and the monodromy group is not or . This proves a conjecture of Guralnick and several conjectures of Guralnick and Shareshian.
Paper Structure (14 sections, 26 theorems, 148 equations, 4 tables)

This paper contains 14 sections, 26 theorems, 148 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Monodromy and ramification
  4. Setup
  5. Reduction to genera of set stabilizers
  6. The ramification types in Theorems \ref{['thm:general']}, \ref{['thm:Sn']}, and \ref{['thm:main']}
  7. Relating set and point stabilizers
  8. Theorem \ref{['thm:main']} when $g_{Y_1}$ is bounded from below
  9. Almost Galois ramification
  10. Castelnuovo's inequality
  11. Ramification data of indecomposable coverings
  12. Proof of Theorem \ref{['thm:main']}
  13. Nonoccurring ramification data
  14. Existence of coverings with ramification as in Table \ref{['table:two-set-stabilizer']}

Key Result

Theorem 1.1

Fix a nonnegative integer $g$. There exists a constant $N_g$ such that every indecomposable covering $f:X \rightarrow\mathbb{P}^1$ of genus $g_X = g$ and degree $n \geq N_g$ satisfies one of the following:

Theorems & Definitions (63)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Theorem 3.1
  • proof : Proof of Theorem \ref{['thm:Sn']} assuming Theorem \ref{['thm:main']}
  • proof : Proof of Theorem \ref{['thm:general']}
  • Lemma 4.1
  • ...and 53 more