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.