Curves and cycles on K3 surfaces (with an appendix by C. Voisin)

The notion of constant cycle curves on K3 surfaces is introduced. These are curves that do not contribute to the Chow group of the ambient K3 surface. Rational curves are the most prominent examples. We show that constant cycle curves behave in some respects like rational curves. For example, using Hodge theory one finds that in each linear system there are at most finitely many such curves of bounded order. Over finite fields, any curve is expected to be a constant cycle curve, whereas over Q this does not hold. The relation to the Bloch-Beilinson conjectures for K3 surfaces over global fields is discussed. Contents 1 Introduction 69 2 Motivation 71 3 Constant cycle curves 74 4 Constant cycle curves on other surfaces 80 5 Finiteness of constant cycle curves of fixed order 81 6 First examples of constant cycle curves 88 7 More examples: fixed curves 90 8 Bitangent correspondence