On maximal green sequence for quivers arising from weighted projective lines

TL;DR

This paper studies the conditions under which quivers from weighted projective lines admit maximal green sequences, showing existence in some cases and non-existence in wild types, advancing understanding in cluster algebra theory.

Contribution

It proves that within the mutation class of a Gabriel quiver from a weighted projective line, some admit maximal green sequences while others do not, depending on the type.

Findings

Existence of maximal green sequences in certain mutation classes.

Non-existence of maximal green sequences for wild type weighted projective lines.

Characterization of when quivers admit maximal green sequences based on the type.

Abstract

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.