Rational curves on fibered varieties

Abstract LetXbe a complex projective variety with log terminal singularities and vanishing augmented irregularity. In this paper, we prove that ifXadmits a relatively minimal genus-one fibration, then it contains a subvariety of codimension one covered by rational curves contracted by the fibration. We then focus on the case of varieties with numerically trivial canonical bundle and we discuss several consequences of this result.