Extremal rational elliptic threefolds

An elliptic fibration is a proper morphism f : X → Y of normal projective varieties whose generic fibre E is a regular curve of genus 1. The Mordell–Weil rank of such a fibration is defined to be the rank of the finitely generated abelian group Pic0 E of degree-0 line bundles on E. In particular, f is called extremal if its Mordell–Weil rank is 0. The simplest nontrivial elliptic fibration is a rational elliptic surface f : X → P1. There is a complete classification of extremal rational elliptic surfaces, due to Miranda–Persson in characteristic 0 [14] and W. Lang in positive characteristic [12, 13]. (See also Cossec– Dolgachev [4, Section 5.6].) The purpose of the present paper is to produce a corresponding classification of a certain class of extremal rational elliptic threefolds.