Correspondences acting on constant cycle curves on K3 surfaces

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 \(\mbox{ccc}(X)\).