Galois covers of singular curves in positive characteristics

Abstract

We study the étale fundamental group of a singular reduced connected curve defined over an algebraically closed field of an arbitrary prime characteristic. It is shown that when the curve is projective, the étale fundamental group is a free product of the étale fundamental group of its normalization with a free finitely generated profinite group whose rank is well determined. A similar result is established for the tame fundamental groups of seminormal affine curves. In the affine case, we provide an Abhyankar-type complete group theoretic classification on which finite groups occur as the Galois groups for Galois étale connected covers over (singular) affine curves. An analogue of the Inertia Conjecture is also posed for certain singular curves.