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Correspondences acting on constant cycle curves on K3 surfaces

Sara Torelli

TL;DR

The paper investigates how correspondences on K3 surfaces act on the group generated by constant cycle curves, introducing loci $Z_n(L)$ of $n$-torsion points in fibers of linear systems and proving that, when these loci have the expected dimension $2$, they induce endomorphisms of $ccc(X)$ compatible with the generalized Bloch conjecture. It provides explicit geometric constructions for $Z_n(L)$ in several degrees (notably degree $2$ via double covers, and degree $4$ via a quartic) and introduces the explicit correspondences $J$ (quartic) and $T$ (degree $6$ complete intersection) to produce nontrivial constant cycle curves beyond rational ones. The results include non-emptiness and dimension statements for $Z_3(H)$ and $Z_4(H)$ in concrete examples, and they illustrate how such correspondences can generate new constant cycle curves from known ones (e.g., conics) while preserving the conjectural filtration on Chow groups. Overall, the work extends the toolbox for studying Chow groups of zero-cycles on K3 surfaces via geometric correspondences tied to torsion phenomena in linear systems and targeted low-degree models, with implications for the easy direction of the generalized Bloch conjecture.

Abstract

Constant cycle curves on a K3 surface $X$ over $\mathbb{C}$ have been introduced by Huybrechts (2014) as curves whose points all define the same class in the Chow group. In this paper we study correspondences $Z \subseteq X\times X$ over $\mathbb{C}$ acting on the group $\mbox{ccc}(X)$ of cycles generated by irreducible constant cycle curves. We construct for any $n\geq 2$ and any very ample line bundle $L$ a locus $Z_n(L)\subseteq X\times X$ of expected dimension $2$, which yields a correspondence that acts on $\mbox{ccc}(X)$, when it has the expected dimension. We provide examples of $Z_n(L)$ for low $n$ and exhibit one correspondence different from $Z_n(L)$ acting on $\mbox{ccc}(X)$.

Correspondences acting on constant cycle curves on K3 surfaces

TL;DR

The paper investigates how correspondences on K3 surfaces act on the group generated by constant cycle curves, introducing loci of -torsion points in fibers of linear systems and proving that, when these loci have the expected dimension , they induce endomorphisms of compatible with the generalized Bloch conjecture. It provides explicit geometric constructions for in several degrees (notably degree via double covers, and degree via a quartic) and introduces the explicit correspondences (quartic) and (degree complete intersection) to produce nontrivial constant cycle curves beyond rational ones. The results include non-emptiness and dimension statements for and in concrete examples, and they illustrate how such correspondences can generate new constant cycle curves from known ones (e.g., conics) while preserving the conjectural filtration on Chow groups. Overall, the work extends the toolbox for studying Chow groups of zero-cycles on K3 surfaces via geometric correspondences tied to torsion phenomena in linear systems and targeted low-degree models, with implications for the easy direction of the generalized Bloch conjecture.

Abstract

Constant cycle curves on a K3 surface over have been introduced by Huybrechts (2014) as curves whose points all define the same class in the Chow group. In this paper we study correspondences over acting on the group of cycles generated by irreducible constant cycle curves. We construct for any and any very ample line bundle a locus of expected dimension , which yields a correspondence that acts on , when it has the expected dimension. We provide examples of for low and exhibit one correspondence different from acting on .
Paper Structure (16 sections, 25 theorems, 28 equations)

This paper contains 16 sections, 25 theorems, 28 equations.

Table of Contents

  1. Introduction.
  2. Preliminaries.
  3. Correspondences on smooth projective surfaces.
  4. Constant cycle curves and correspondences on K3 surfaces.
  5. Generalized Bloch conjecture.
  6. Correspondences from $n$-torsion points of linear systems on $K3$ surfaces.
  7. The loci $Z'_n(L)$ and $Z_n(L)$.
  8. Properties of the correspondence $Z_n(L)$.
  9. Correspondences on K3 surfaces of degree $2$.
  10. Correspondences on K3 surfaces of degree $4$.
  11. Construction and properties of the correspondence $J$.
  12. Non-emptiness of $Z_3(H)$ defined by a hyperplane section.
  13. The correspondence $J$ and constant cycle curves.
  14. Correspondences on K3 surfaces of degree $6$.
  15. Construction of the correspondence $T$.
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $X$ be a smooth projective K3 surface over ${\mathbb C}$ and $L$ a very ample line bundle. Then for any integer $n\geq 2$ the locus $Z_n(L)\subseteq X\times X$ defined as the closure of is a locus of expected dimension $2$. Moreover, $Z_n(L)$ acts on $\mathop{\mathrm{ccc}}\nolimits(X)$ when it has dimension $2$ (see Definition def:corrccc).

Theorems & Definitions (56)

  • Theorem 1.1
  • Conjecture 1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • ...and 46 more