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Monodromy groups of product type

Danny Neftin, Michael E. Zieve

Abstract

The combination of this paper and its companion complete the classification of monodromy groups of indecomposable coverings of complex curves $f:X\rightarrow \mathbb P^1$ of sufficiently large degree in comparison to the genus of $X$. In this paper we determine all such coverings with monodromy group $G\leq S_\ell\wr S_t$ of product type for $t\ge 2$.

Monodromy groups of product type

Abstract

The combination of this paper and its companion complete the classification of monodromy groups of indecomposable coverings of complex curves of sufficiently large degree in comparison to the genus of . In this paper we determine all such coverings with monodromy group of product type for .
Paper Structure (20 sections, 32 theorems, 143 equations, 5 tables)

This paper contains 20 sections, 32 theorems, 143 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Monodromy
  4. Ramification
  5. Riemann--Hurwitz
  6. Riemann's existence theorem
  7. Wreath products and reduced forms
  8. Groups of product type
  9. Setup
  10. The ramification types in Theorem \ref{['thm:main-wreath']}
  11. Almost Galois ramification
  12. The genus of a product type covering
  13. Ramification of genus $\leq 1$ with a transposition
  14. Theorem \ref{['thm:main-wreath']}: reduction to almost Galois types and the case $t\geq 3$
  15. Theorem \ref{['thm:main-Gal']}: The case $t=2$
  16. ...and 5 more sections

Key Result

Theorem 1.2

Fix an integer $t\geq 2$. There exist positive constants $c=c_t, d=d_t$ depending only on $t$, such that for every indecomposable covering $f:X\rightarrow X_0$ with monodromy group $G$ of product type acting on $\Delta^t$, either $g_X > c\ell^{t-1}-d\ell^{t-2}$ or the ramification type of $f$ appear

Theorems & Definitions (82)

  • Conjecture 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Remark 2.3
  • Lemma 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • ...and 72 more