Extend the tiling-algebra realization to other surfaces

Establish an analogue of the isomorphism between the Hom-Ext quiver algebra and the tiling algebra for exceptional arc diagrams on surfaces other than the annulus, including appropriately weaker hypotheses in non-hereditary cases.

Background

The paper identifies the Hom-Ext quiver of an exceptional collection for a quiver of type $\widetilde{\mathbb A}$ with the tiling algebra of the corresponding exceptional arc diagram on an annulus. The authors ask whether this geometric-algebraic correspondence persists on other surfaces.

The question is motivated by the fact that the tiling-algebra construction in the cited literature assumes that the arcs form a partial triangulation, whereas the paper suggests that Hom-Ext quivers may extend the construction to collections of arcs with more general intersection behavior. The authors specifically anticipate that non-hereditary cases may require weaker assumptions on the collection.