Galois Covers of Singular Curves in Positive Characteristics

Abstract

We study the étale fundamental groups of singular reduced connected curves defined over an algebraically closed field of 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. As a consequence of this result and the known results for the smooth case, necessary conditions are given for a finite group to appear as a quotient of the étale fundamental group. Next, we provide similar results for an affine integral curve $U$. We provide a complete group theoretic classification on which finite groups occur as the Galois groups for Galois étale connected covers of $U$. In fact, when $U$ is a seminormal curve embedded in a connected seminormal curve $X$ such that $X - U$ consists of smooth points, the tame fundamental group $π_1^t(U \subset X)$ is shown to be isomorphic to a free product of the tame fundamental group of the normalization of $U$ with a free finitely generated profinite group whose rank is known. An analogue of the Inertia Conjecture is also posed for certain singular curves.