Elliptic constant cycle curves on Kummer surfaces

TL;DR

This paper computes the order of elliptic constant cycle curves on Kummer surfaces using the transcendental intermediate Jacobian, showing that any positive integer can be realized as such an order on a K3 surface.

Contribution

It provides a method to determine the order of elliptic constant cycle curves on Kummer surfaces and demonstrates that all positive integers can be realized as these orders.

Findings

The order of elliptic constant cycle curves can be computed via the transcendental intermediate Jacobian.

Every positive integer can be realized as the order of a constant cycle curve on a K3 surface.

The paper establishes a link between the order of these curves and their geometric origin on Kummer surfaces.

Abstract

The order of a constant cycle curve $C \subset X$ on a K3 surface, defined by Huybrechts, is a positive integer that measures the obstruction to decomposing the diagonal class $\Delta_C$ in the Chow group $\mathrm{CH}^2(X \times C)$. In this paper, we compute the order of elliptic constant cycle curves that naturally arise on Kummer surfaces, by passing to the transcendental intermediate Jacobian $J_{\mathrm{tr}}^3(X \times C)$. As a consequence, every $n \in \mathbb{N}$ can be realized as the order of a constant cycle curve on a K3 surface.