# Kummer Coverings

Let $L_1$, $\cdots$, $L_k$ be homogenous linear forms in three variables $z_0$, $z_1$, $z_2$ defining $k$ lines in $\mathbb{P}_{2}$. Consider the abelian extension $K ((L_2/L_1)^{1/n}, \cdots, (L_k/L_1)^{1/n}) \supset K:= \mathbb{C}(z_{1}/z_{0}, z_{2}/z_{0})$ with group $G:= (\mathbb{Z}/n\mathbb{Z})^{k-1}$. According to the book Barth-Hulek-Peters-Vandeven (Compact Complex Surfaces, page 240-42), such extension corresponds to a covering $f: X \rightarrow \tilde{\mathbb{P}_{2}}$. Let $P$ be one of the points where at least three lines in $\mathbb{P}_{2}$ meet. Then the lines through point $P$ defines a fibration. I hope to understand the fibration structure. What are the singular fibers and what is the genus of this fibration? If it is any helpful, you may consider the configurations $A_{1}(6)$ (complete quadrangle). Any reference on this is greatly appreciated.

The answer depends on the integer $n$, on the number $k$ of lines and on the configuration of lines. In section 2.5 of this paper
you can find a computation for $n=5$ and $A_1(6)$.