Finite presentation of the tame fundamental group

Abstract

Let p be a prime number, and let k be an algebraically closed field of characteristic p. We show that the tame fundamental group of a smooth affine curve over k is a projective profinite group. We prove that the fundamental group of a smooth projective variety over k is finitely presented; more generally, the tame fundamental group of a smooth quasi-projective variety over k, which admits a good compactification, is finitely presented.