Embedding problems for the ´etale fundamental group of curves

Abstract

Let X be a smooth projective curve over an algebraically closed field k of char- acteristic p > 0, S be a finite subset of closed points in X. Given an embedding problem (β : Γ ↠ G, α : π´et 1 (X \S) ↠ G) for the ´etale fundamental group π´et 1 (X \S), where H = ker(β) is prime-to-p, we discuss when an H-cover W → V of the G- cover V → X corresponding to α is a proper solution. When H is abelian and G is a p-group, some necessary and sufficient conditions for solving the embedding prob- lems are given in terms of the action of G on a certain generalization of Pic0(V )[m], the m-torsion of the Picard group. When a solution exists, we discuss the problem of finding the number of (non-equivalent) solutions and the minimum of genera of the covers corresponding to proper solutions for the given embedding problem.