Rational curves of minimal degree and characterizations of projective spaces

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A survey on the Campana-Peternell Conjecture (2014, background): “By assumption πlm−1 ◦f l[1] does not contract fibers of p2, since their image in X is numerically equivalent to Γ, and it is generically finite when restricted to fibers of p1, by the choice of l.”
On Fano foliations (2011, unknown): “By [2], after shrinking X0 and T0 if necessary, π0 : X0 → T0 becomes a P2-bundle over a smooth base.”
Cohomological characterizations of projective spaces and hyperquadrics (2007, background): “In [ Ara06 , Proposition 4.1] D is assumed to be ample, but the proof only uses the fact that f ∗D is a subsheaf of f ∗T +”