24 rational curves on K3 surfaces

Abstract

Given [Formula: see text], we prove that all smooth K3 surfaces (over any field of characteristic [Formula: see text]) of degree greater than [Formula: see text] contain at most 24 rational curves of degree at most [Formula: see text]. In the exceptional characteristics, the same bounds hold for non-unirational K3 surfaces, and we develop analogous results in the unirational case. For [Formula: see text], we also construct K3 surfaces of any degree greater than [Formula: see text] with 24 rational curves of degree exactly [Formula: see text], thus attaining the above bounds.