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Let $\Sigma$ be a smooth projective surface, let $f' : S' \to \Sigma$ be adouble 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 asplitting 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 setof $\mu$. In this article, we show that a kind of "reciprocity" of splittingcurves holds for a certain pair of curves on rational ruled surfaces. As anapplication, we consider the topology of the complements of certain curves onrational ruled surfaces.

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