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Let $X$ be a complex smooth Fano variety whose minimal anticanonical degreeof non-free rational curves on $X$ is at least $\dim X-2$. We give aclassification of extremal contractions of such varieties. As applications, weobtain a classification of Fano fourfolds whose pseudoindex and Picard numberare greater than one and study the structure of Fano varieties with nef thirdexterior power of the tangent bundle.

Extremal contractions and non-free rational curves on Fano varieties