THE COMBINATORICS OF DEGENERATE COVERS AND AN APPLICATION FOR GENERAL CURVES OF GENUS 3

Abstract

Let $C_g$ be a general curve of genus $g$. If $g\ge 4$ then the monodromy group of a primitive cover $C_g\to {\mathrm{\mathbb{P}}}^1$ of degree $n$ is either $S_n$ or $A_n$, and both cases actually occur (under suitable conditions on $n$ for fixed $g$). For $g=3$ also the groups $G{L}_3\left(2\right)$ and $AG{L}_3\left(2\right)$ occur. In the present paper we settle the last possible case of $AG{L}_4\left(2\right)$. This requires new methods (which may be of independent interest) studying the combinatorial structure of degenerate covers.