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 $X\rightarrow B$ does contain vertical rational curves if and only if it not isomorphic to a finite \'etale quotient of a product $\tilde{B}\times E$ 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.