Rational curves on genus one fibrations

TL;DR

This paper investigates conditions under which genus one fibrations contain rational curves, providing necessary and sufficient criteria and establishing various sufficient conditions for their existence.

Contribution

It offers new criteria for the presence of rational curves in genus one fibrations, including a characterization involving finite étale quotients and vertical rational curves.

Findings

Vertical rational curves exist iff the variety is not a finite étale quotient of a product.

Several sufficient conditions for rational curves in genus one fibrations are established.

The paper characterizes when a genus one fibration admits rational curves based on singularity and quotient conditions.

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.