Rational curves on K3 surfaces (joint with F. Bogomolov and Y. Tschinkel) We propose an extension of the Mori-Mukai technique for constructing rational curves on projective K3 surfaces. While their approach shows that `general' K3 surfaces admit infinitely many rational curves, it leaves open the possibility that a specific K3 surface might only have a finite number. Essentially, we reduce mod p, analyze the rational curves on the resulting surface over a finite field, and lift these to characteristic zero. As an application, we show that K3 surfaces whose Picard group is generated by a degree-two line bundle always admit an infinite number of rational curves.