A note on quadratic residue curves on rational ruled surfaces

Abstract

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 of $S'$. Put $f = f'\circ\mu$. An irreducible curve $D$ on $\Sigma$ is said to be a splitting curve with respect to $f$ if $f^*D$ is of the form $D^+ + D^- + E$, where $D^+ \neq D^-$, $D^- = \sigma_f^* D^+$, $\sigma_f$ being the covering transformation of $f$ and all irreducible components of $E$ are contained in the exceptional set of $\mu$. In this article, we consider "reciprocity" concerning splitting curves when $\Sigma$ is a rational ruled surface.