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 we prove that the tame fundamental group of a smooth quasi-projective variety over $k$ which admits a good compactification is finitely presented. v2: references added. Thank you to all for the friendly and fruitful comments.