Constructing rational curves on K3 surfaces

TL;DR

This paper introduces a new method combining reduction modulo p and lifting techniques to construct rational curves on K3 surfaces, proving the existence of infinitely many such curves on certain complex K3 surfaces.

Contribution

It develops a mixed-characteristic approach to produce rational curves on K3 surfaces, extending the Mori-Mukai technique to new settings.

Findings

Proves all Picard rank 1 K3 surfaces with degree two have infinitely many rational curves.

Introduces a reduction and lifting method for constructing rational curves.

Establishes the existence of rational curves on complex K3 surfaces with specific Picard groups.

Abstract

We develop a mixed-characteristic version of the Mori-Mukai technique for producing rational curves on K3 surfaces. We reduce modulo p, produce rational curves on the resulting K3 surface over a finite field, and lift to characteristic zero. As an application, we prove that all complex K3 surfaces with Picard group generated by a class of degree two have an infinite number of rational curves.