Generic cluster characters

Let $\text{CC}$ be a Hom-finite triangulated 2-Calabi-Yau category with a cluster-tilting object $T$. Under a constructibility condition we prove the existence of a set ${\mathcal{G}}^T(\text{CC})$ of generic values of the cluster character associated to $T$. If $\text{CC}$ has a cluster structure in the sense of Buan-Iyama-Reiten-Scott, ${\mathcal{G}}^T(\text{CC})$ contains the set of cluster monomials of the corresponding cluster algebra. Moreover, these sets coincide if $\mathcal{C}$ has finitely many indecomposable objects. When $\text{CC}$ is the cluster category of an acyclic quiver and $T$ is the canonical cluster-tilting object, this set coincides with the set of generic variables previously introduced by the author in the context of acyclic cluster algebras. In particular, it allows to construct $\mathbb{Z}$-linear bases in acyclic cluster algebras.