Splitting curves on a rational ruled surface, the Mordell-Weil groups of hyperelliptic fibrations and Zariski pairs

Let $\Sigma$ be a smooth projective surface, let $f' : S' \to \Sigma$ be a double cover of $\Sigma$ and let $\mu : S \to S'$ be the canonical resolution. Put $f = f'\circ\mu$. An irreducible curve $C$ on $\Sigma$ is said to be a splitting curve with respect to $f$ if $f^*C$ is of the form $C^+ + C^- + E$, where $C^- = \sigma_f^*C^+$, $\sigma_f$ being the covering transformation of $f$ and all irreducible components of $E$ are contained in the exceptional set of $\mu$. In this article, we show that a kind of "reciprocity" of splitting curves holds for a certain pair of curves on rational ruled surfaces. As an application, we consider the topology of the complements of certain curves on rational ruled surfaces.