Rational Curves on K3 Surfaces

We prove that complex projective K3 surfaces of odd Picard rank contain infinitely many rational curves. The first step, due to Bogomolov, Hassett, and Tschinkel, is to reduce to positive characteistic, where existence of rational curves is established using the Weil conjectures (proven by Deligne) and the Artin-Tate conjecture (in the most important cases proven by Nygaard, and Nygaard-Ogus). The second step is to use the moduli space of genus-zero stable maps to lift to characteristic zero. To overcome the difficulties of this lifting problem, we introduce a class of genus-zero stable maps that we call "rigidifiers". This is joint work with Jun Li.