On maximal green sequence for quivers arising from weighted projective lines

We investigate the existence and non-existence of maximal green sequences for quivers arising from weighted projective lines. Let $Q$ be the Gabreil quiver of the endomorphism algebra of a basic cluster-tilting object in the cluster category $mathcal{C}_mathbb{X}$ of a weighted projective line $mathbb{X}$. It is proved that there exists a quiver $Q$ in the mutation equivalence class $operatorname{Mut}(Q)$ such that $Q$ admits a maximal green sequence. On the other hand, there is a quiver in $operatorname{Mut}(Q)$ which does not admit a maximal green sequence if and only if $mathbb{X}$ is of wild type.