The following theorem was proved in [4] over the complex numbers. It turns out that the proof given there works with very small adjustments in arbitrary characteristic.
CITATION STYLE
Kovács, S. J. (2013). The cone of curves of K3 surfaces revisited. In Birational Geometry, Rational Curves, and Arithmetic (pp. 163–169). Springer New York. https://doi.org/10.1007/978-1-4614-6482-2_8
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