Elliptic constant cycle curves 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 $\hat{I}{”}_C$ 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.