Periodic sharpness of Miyaoka's bound for smooth rational curves

TL;DR

This paper investigates the maximum number of smooth rational curves of a given degree on complex K3-surfaces, providing precise conditions under which Miyaoka's bound is sharp for large n and specific degrees.

Contribution

It offers a detailed characterization of configurations of rational curves on K3-surfaces where Miyaoka's bound is attained, especially for large n and odd nd.

Findings

Maximal number of rational degree d curves on K3-surfaces determined

Conditions for sharpness of Miyaoka's bound characterized

Configurations of rational curves explicitly described

Abstract

We determine the maximal number of smooth rational degree d curves on a complex K3-surface of degree 2n provided n is sufficiently large as compared to d>1. We obtain precise characterization of configurations of rational degree d curves for which Miyaoka's bound is sharp for nd odd and n sufficiently large as compared to d.