Rational curves on genus-one fibrations

Abstract

In this paper we look for necessary and sufficient conditions for a genus-one fibration to have rational curves. We show that a projective variety with log terminal singularities that admits a relatively minimal genus-one fibration $$X\rightarrow B$$ X → B does contain vertical rational curves if and only if it not isomorphic to a finite étale quotient of a product $$\tilde{B}\times E$$ B ~ × E over B. Many sufficient conditions for the existence of rational curves in a variety that admits a genus-one fibration are proved in this paper.