Erratum to: Normal subgroups of the algebraic fundamental group of affine curves in positive characteristic

Abstract

Let k be an algebraically closed field of characteristic p > 0 of countable cardinality. Denote by C a smooth, connected, affine curve defined over k with function field K and by π1(C) its algebraic fundamental group. In the original paper we addressed the question of the structure of the normal closed subgroups of π1(C). More precisely, in Theorem 1.1, we gave a sufficient condition for such a normal closed subgroup N to be isomorphic to a subgroup of a free profinite group of countable rank. There we claimed to prove this under the hypothesis that the group M = π1(C)/N had an infinitely generated Sylow p-subgroup. However, as remarked in the paragraph after Corollary 1.2 of [1], this hypothesis is not enough. We would like to make this remark more explicit by stating (as noticed by those authors) the additional hypothesis needed to make Theorem 1.1 work. For each integer g ≥ 0 denote by Pg(C) the intersection of algebraic fundamental groups