On Galois Coverings of curves and their Families

Abstract

In this paper, we describe Galois covers of algebraic curves and their families by using local systems associated to push-forward of sheaves by the structure morphism. More precisely, if $f:C\to Y$, we consider the sheaves $f_*(\C)$. The group action by the Galois group $G$, yields a decomposition of this sheaf into irreducible local systems corresponding to irreducible representations of the group $G$. If $\rho$ is such an irreducible representation, the eigensheaf $\L_{\rho}$ of $f_*(\C)$ gives rise to another useful sheaf which is related to the homology group $H_1(C,\C)$. Using this, we describe the action of the Galois group $G$ on the homology group. As a particular example, we study the Dihedral covers of $\P^1$ in some detail.