The Picard number of fibred Mori dream surfaces

Abstract

Let $S$ be a smooth complex projective surface endowed with a fibration $f \colon S \to C$ onto a smooth projective curve $C$. We prove that, if $S$ is a Mori dream space (or, more generally, if its pseudo-effective cone is polyhedral) then the Picard number $ρ(S)$ can be effectively computed by counting the irreducible components of the reducible fibres of $f$. A first simple consequence is that, given an elliptic fibration $f \colon S \to \mathbb{P}^1$ with a section and such that $S$ is a Mori dream space, the Mordell-Weil group of the general fibre of $f$ is finite. The main application is a simple criterion for proving that a surface fibred over a curve is not a Mori dream space. We show that certain Horikawa surfaces, Fermat surfaces in $\mathbb{P}^{3}$ of every degree $\ge 4$, particular product-quotient surfaces, and the minimal simply connected numerical Godeaux surface constructed by Craighero and Gattazzo are not Mori dream spaces.