A note on splitting curves of plane quartics and multi-sections of rational elliptic surfaces

Abstract

In this note, we study the relation between splitting curves of plane quartics and multi-sections of rational elliptic surfaces associated to the quartics. We give a criterion for when a simple contact curve splits and calculate the images of the components of the corresponding multi-sections in the Mordell–Weil group of the associated rational elliptic surface. We also introduce a notion called the splitting type for certain configurations of plane curves which allows us to distinguish the configurations topologically. To demonstrate its effectiveness we construct new examples of Zariski pairs and a Zariski triple.