Constructing rational curves on K3 surfaces

Abstract

We develop a mixed-characteristic version of the Mori-Mukai technique for producing rational curves on K$3$ surfaces. We reduce modulo $p$, produce rational curves on the resulting K$3$ surface over a finite field, and lift to characteristic zero. As an application, we prove that all complex K$3$ surfaces with Picard group generated by a class of degree two have an infinite number of rational curves.