Galois Covers of Singular Curves in Positive Characteristics

TL;DR

This paper investigates the structure of the étale fundamental groups of singular algebraic curves over algebraically closed fields of prime characteristic, providing new classifications and conditions for Galois covers, especially in the affine and seminormal cases.

Contribution

It offers a novel group-theoretic classification of finite groups appearing as Galois groups of covers of singular curves in positive characteristic, extending known results from smooth to singular cases.

Findings

Fundamental group of projective singular curves is a free product involving the normalization.

Complete classification of Galois groups for covers of affine integral curves.

Isomorphism of tame fundamental groups with free products in seminormal curve cases.

Abstract

We study the \'{e}tale 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 \'{e}tale fundamental group is a free product of the \'{e}tale 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 \'{e}tale 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 \'{e}tale 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 $\pi_1^t(U \subset X)$ 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.