Curves and cycles on K3 surfaces

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. E.g. 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 number fields this does not hold. The relation to the Bloch--Beilinson conjectures for K3 surfaces over global fields is discussed.