Rational Curves on K3 Surfaces

TL;DR

This paper proves that certain K3 surfaces with odd Picard rank have infinitely many rational curves, extending existing methods by using moduli spaces and positive characteristic techniques.

Contribution

It introduces an extension of the Bogomolov-Hassett-Tschinkel approach to demonstrate the abundance of rational curves on specific K3 surfaces.

Findings

Infinitely many rational curves on K3 surfaces with odd Picard rank

Extension of moduli space techniques to positive characteristic

Advancement in understanding rational curves on algebraic surfaces

Abstract

We show that projective K3 surfaces with odd Picard rank contain infinitely many rational curves. Our proof extends the Bogomolov-Hassett-Tschinkel approach, i.e., uses moduli spaces of stable maps and reduction to positive characteristic.