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Curves and cycles on K3 surfaces

Daniel Huybrechts, Claire Voisin

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.

Curves and cycles on K3 surfaces

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.

Paper Structure

This paper contains 11 sections, 38 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Constant cycle curves
  4. Constant cycle curves on other surfaces
  5. Finiteness of constant cycle curves of fixed order
  6. First examples of constant cycle curves
  7. More examples: fixed curves
  8. Bitangent correspondence
  9. Finite fields
  10. Further questions and remarks
  11. Appendix by Claire Voisin

Key Result

Lemma 3.4

Let $X$ be a K3 surface over an algebraically closed field $k$. For an integral curve $C\subset X$ the following conditions are equivalent: i) The curve $C$ is a constant cycle curve. ii) There exists a positive integer $n$ such that in ${\rm CH}^2(X\times_k{k(\eta_C)})$, where the generic point $\eta_C\in C$ is viewed as a closed point in $X\times_k{k(\eta_C)}$. iii) If $\eta_C\in C$ is viewed a

Theorems & Definitions (87)

  • Conjecture 2.1
  • Conjecture 2.2
  • Conjecture 2.3
  • Conjecture 2.4
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Definition 3.5
  • ...and 77 more