Rational curves on fibered varieties

TL;DR

This paper proves that certain fibered varieties with specific singularities and trivial canonical bundle contain subvarieties covered by rational curves, revealing new geometric structures and implications.

Contribution

It establishes the existence of rationally covered subvarieties in fibered varieties with log terminal singularities and trivial canonical bundle, extending understanding of their geometry.

Findings

Existence of rational curves in fibered varieties with log terminal singularities.

Identification of subvarieties covered by rational curves contracted by the fibration.

Implications for varieties with numerically trivial canonical bundle.

Abstract

Let $X$ be a projective variety with log terminal singularities and vanishing augmented irregularity. In this paper we prove that if $X$ admits a relatively minimal genus one fibration then it does contain 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.