Curves and cycles on K3 surfaces

TL;DR

This paper introduces constant cycle curves on K3 surfaces, explores their properties, and discusses their behavior over different fields, connecting to deep conjectures in algebraic geometry.

Contribution

It defines constant cycle curves on K3 surfaces and analyzes their properties, including finiteness results and field-dependent behaviors, linking to Bloch--Beilinson conjectures.

Findings

Finitely many constant cycle curves of bounded order in each linear system.

Over finite fields, all curves are expected to be constant cycle curves.

The behavior over number fields differs significantly from finite fields.

Abstract

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.