# 24 rational curves on K3 surfaces

## TL;DR

This paper establishes upper bounds on the number of rational curves of bounded degree on K3 surfaces over various fields, and constructs examples achieving these bounds, advancing understanding of rational curves on K3 surfaces.

## Contribution

It proves bounds on rational curves on K3 surfaces over different fields and constructs explicit examples reaching these bounds.

## Key findings

- Bound of 24 rational curves of degree at most d on K3 surfaces of degree > 84d^2
- Bounds hold in all characteristics except 2,3, with adaptations for non-unirational and unirational cases
- Explicit constructions of K3 surfaces with exactly 24 rational curves of degree d

## Abstract

Given d in IN, we prove that all smooth K3 surfaces (over any field of characteristic p other than 2,3) of degree greater than 84d^2 contain at most 24 rational curves of degree at most d. In the exceptional characteristics, the same bounds hold for non-unirational K3 surfaces, and we develop analogous results in the unirational case. For d at least 3, we also construct K3 surfaces of any degree greater than 4d(d+1) with 24 rational curves of degree exactly d, thus attaining the above bounds.