A more precise form of this result is originally due to Miyaoka and Mori, see Theorem 3.6 page 67 in 

<cite authors="Debarre, Olivier">*Olivier Debarre*: Higher-dimensional algebraic geometry, Universitext. New York, NY: Springer. xiii, 233 p. (2001). [ZBL0978.14001](https://zbmath.org/?q=an:0978.14001).</cite>

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