We develop a mixed-characteristic version of the Mori-Mukai technique for producing rational curves on K surfaces. We reduce modulo , produce rational curves on the resulting K surface over a finite field, and lift to characteristic zero. As an application, we prove that all complex K surfaces with Picard group generated by a class of degree two have an infinite number of rational curves.
Fedor Bogomolov. Brendan Hassett. Yuri Tschinkel. "Constructing rational curves on K3 surfaces." Duke Math. J. 157 (3) 535 - 550, 15 April 2011. https://doi.org/10.1215/00127094-1272930