Low-genus primitive monodromy groups with a nonunique minimal normal subgroup

TL;DR

This paper classifies low-genus Riemann surface coverings with primitive monodromy groups having multiple minimal normal subgroups, showing uniqueness for genus 0 and 1, and nonexistence for large degrees at any genus.

Contribution

It proves the uniqueness of such coverings for genus 0 and 1, and establishes nonexistence for large degrees at any genus, filling gaps in the classification of monodromy groups.

Findings

Only one such covering exists for genus ≤ 1.

No such coverings exist for large degree regardless of genus.

Clarifies the structure of primitive monodromy groups with multiple minimal normal subgroups.

Abstract

Let $X$ be a Riemann surface, and let $f:X\to\mathbb{P}^1_\mathbb{C}$ be an indecomposable (branched) covering of genus $g$ and degree $n$ whose monodromy group has more than one minimal normal subgroup. Closing a gap in the literature, we show that there is only one such covering when $g\leq 1$. Moreover, for arbitrary $g$, there are no such coverings with $n\gg_g 0$ sufficiently large.