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 Gabriel quiver of the endomorphism algebra of a basic cluster-tilting object in the cluster category C X of a weighted projective line X . It is proved that there exists a quiver Q ′ in the mutation equivalence class Mut( Q ) of Q such that Q ′ admits a maximal green sequence. Furthermore, there is a quiver in Mut( Q ) which does not admit a maximal green sequence if and only if X is of wild type.