On Galois Coverings of curves and their Families

Abolfazl Mohajer

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arXiv · 2304.12883

On Galois Coverings of curves and their Families

Abolfazl Mohajer

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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.

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Abolfazl Mohajer. 2023-04-25. On Galois Coverings of curves and their Families. https://arxiv.org/abs/2304.12883

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