Extremal hyperelliptic fibrations on rational surfaces

TL;DR

This paper studies extremal hyperelliptic fibrations on rational surfaces, characterizing those with maximal Picard number and trivial Mordell-Weil group, and provides explicit defining equations.

Contribution

It characterizes extremal hyperelliptic fibrations with maximal Picard number and trivial Mordell-Weil group on rational surfaces, including explicit equations.

Findings

Picard number is bounded by the genus of the fiber.

Maximal Picard number corresponds to specific singular fibers.

Explicit equations are derived for extremal cases.

Abstract

We consider a rational surface with a relatively minimal fibration. Picard number of a such fibred surface is bounded in terms of the genus of a general fibre. When Picard number is the maximum for any given genus, we characterize a such fibred surface whose Mordell-Weil group is trivial by singular fibres. Furthermore, we describe the defining equation explicitly.