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

Abstract

Let $\Sigma$ be a smooth projective surface, let $f^{\prime }:S^{\prime }\to \Sigma$ be a double cover of $\Sigma$ and let $\mu :S\to S^{\prime }$ be the canonical resolution. Put $f=f^{\prime }\circ \mu$. An irreducible curve $C$ on $\Sigma$ is said to be a splitting curve with respect to $f$ if $f^{\ast }C$ is of the form $C^{+}+C^{-}+E$, where $C^{-}={\sigma }_f^{\ast }{C}^{+}$, ${\sigma }_f$ 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.

23pages

References in corpus (1)

• Lattice Zariski k-ples of plane sextic curves and Z-splitting curves for double plane sextics