Note sur la détermination algébrique du groupe fondamental pro-résoluble d'une courbe affine

Abstract

Let X be a smooth projective algebraic curve of genus g minus $r\ge 1$ points defined over an algebraically closed field k of characteristic $p\ge 0$. The structure of the largest prime to p quotient of the étale fundamental group is well known by transcendental methods : it is isomorphic to the largest prime to p quotient of a free pro-finite group on 2g+r-1 generators. We show that, with purely algebraic means, we can prove the corresponding result for the largest pro-solvable quotient of these groups.

v2, minor changes, exposition improved