Monodromy groups of indecomposable coverings of bounded genus

TL;DR

This paper classifies the ramification types and monodromy groups of indecomposable complex curve coverings of genus g, confirming several conjectures of Guralnick and Shareshian for large degree n.

Contribution

It provides a comprehensive classification of monodromy groups for indecomposable coverings of bounded genus, verifying key conjectures in the field.

Findings

Classification of ramification types for large degree n

Identification of monodromy groups not equal to A_n or S_n

Proof of conjectures by Guralnick and Shareshian

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 $n:=\deg(f)$ 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.