Extremal trigonal fibrations on rational surfaces

Abstract

For a relatively minimal fibration $f : X \to \mathbb{P}^1$ of non-hyperelliptic curves of genus $g$, we know the Picard number $\rho(X) \leq 3g + 8$. We study the case where $\rho(X) = 3g + 8$ and the Mordell–Weil group of $f$ is trivial. Such an $f$ occurs only if $g \equiv 0$ or $1 \pmod{3}$, and we describe such $f : X \to \mathbb{P}^1$ explicitly.