On cyclic groups covers of the projective line

Abstract

This article extends the study of cyclic ramified covers of the projective line defined by Kummer equations. We consider the most general case of such covers, allowing arbitrary orders in the roots of the generating radicant. The primary goal is the computation of the fundamental group of both the open and complete curve. We employ tools of combinatorial group theory utilizing the Smith Normal Form. This result is further visualized through the theory of foldings and $S$-graphs. Finally, we apply the theory of Alexander modules and the Crowell exact sequence to compute the abelianization of the fundamental group, $H_{1}(X, \mathbb{Z})$, and determine its Galois~module~structure over a field $k$ confirming the result using the Chevalley-Weil formula.