A geometric model for the module category of a skew-gentle algebra (Q6038739)

Let \(\mathbf{S}\) be a punctured marked surface, that is, a compact oriented surface with a finite set of marked points on its non-empty boundary and with a finite set of punctures in its interior. To any admissible partial triangulation \(\mathbf{T}\) (Definition 1.8) of \(\mathbf{S}\), the authors of the paper construct an associated algebra denoted by \(\Lambda^\mathbf{T}\) and called a skew-tiling algebra (Definition 1.9). They prove that any skew-tiling algebra is a skew-gentle algebra and conversely, for any skew-gentle algebra \(A\), there exists a punctured marked surface \(\mathbf{S}\) with an admissible partial triangulation \(\mathbf{T}\) such that \(\Lambda^\mathbf{T}\cong A\). Based on this, the authors establish a bijection between tagged permissible curves and certain indecomposable modules, interpret the Auslander-Reiten translation via the tagged rotation, and show intersection-dimension formulas. Moreover, as an application, they classify support \(\tau\)-tilting modules via maximal collections of non-crossing tagged generalized permissible curves.