Normal subgroups of fundamental groups of affine curves in positive characteristic II

TL;DR

This paper characterizes specific closed normal subgroups of the algebraic fundamental group of affine curves over algebraically closed fields of positive characteristic, extending previous results to uncountable fields.

Contribution

It generalizes earlier work by removing the countability restriction on the base field, providing a broader characterization of normal subgroups in this setting.

Findings

Identifies a subset of closed normal subgroups of $\

Extends previous results to algebraically closed fields of arbitrary cardinality.

Provides a detailed description of the structure of these normal subgroups.

Abstract

Let $k$ be an algebraically closed field of characteristic $p>0$ and let $C/k$ be a smooth connected affine curve. Denote by $\pi_1(C)$ its algebraic fundamental group. The goal of this paper is to characterize a certain subset of closed normal subgroups $N$ of $\pi_1(C)$. In "Normal subgroups of fundamental groups of affine curves in positive characteristic" we proved the same result under the additional hypothesis that $k$ had countable cardinality.