The fundamental group of affine curves in positive characteristic

TL;DR

This paper proves that the commutator subgroup of the fundamental group of a smooth affine curve over an uncountable algebraically closed field of positive characteristic is a free profinite group with rank equal to the field's cardinality.

Contribution

It establishes a new structural understanding of the fundamental group in positive characteristic, showing its commutator subgroup is a free profinite group of large rank.

Findings

The commutator subgroup is a profinite free group.

Rank of the commutator subgroup equals the cardinality of the base field.

Results apply to smooth affine curves over uncountable algebraically closed fields.

Abstract

It is shown that the commutator subgroup of the fundamental group of a smooth affine curve over an uncountable algebraically closed field $k$ of positive characteristic is a profinite free group of rank equal to the cardinality of $k$.