Pencils of genus two curves on rational surfaces

TL;DR

This paper classifies certain genus two curve fibrations on rational surfaces, identifying only four canonical models and providing examples with trivial Mordell-Weil groups, advancing understanding of their geometric structures.

Contribution

It introduces a classification of genus two fibrations on rational surfaces, identifying canonical models and examples with trivial Mordell-Weil groups.

Findings

Only four canonical pencils exist among possible models.

Canonical pencils can have trivial Mordell-Weil groups.

Fibrations are interpreted via birational morphisms as pencils of plane curves.

Abstract

We consider relatively minimal fibrations of curves of genus two on rational surfaces whose Picard numbers are not maximal. By birational morphisms, such fibred surfaces are interpreted as pencils of plane curves. We show that only four are canonical, among a variety of possible models. For each canonical pencil, we give an example with trivial Mordell-Weil group.