PRENDERGAST-SMITH, A., 2010. Extremal rational elliptic threefolds. Michigan Mathematical Journal, 59 (3), pp. 535 - 572
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  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.
This article was published in Michigan Mathematical Journal and is available here with the kind permission of the publisher..