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A well known result of Miyaoka asserts that a complex projective manifold is uniruled if its cotangent bundle restricted to a general complete intersection curve is not nef. Using the Harder-Narasimhan filtration of the tangent bundle, it can moreover be shown that the choice of such a curve gives rise to a rationally connected foliation of the manifold. In this note we show that, conversely, a movable curve can be found so that the maximal rationally connected fibration of the manifold may be recovered...

In this paper we classify rank two Fano bundles $ℰ$ on Fano manifolds satisfying $H^2(X,\mathbb{Z})\cong H^4(X,\mathbb{Z})\cong \mathbb{Z}$. The classification is obtained via the computation of the nef and pseudoeffective cones of the projectivization $\mathbb{P}\left(ℰ\right)$, that allows us to obtain the cohomological invariants of $X$ and $ℰ$. As a by-product we discuss Fano bundles associated to congruences of lines, showing that their varieties of minimal rational tangents may have several linear components.