Schemes of modules over gentle algebras and laminations of surfaces

We study the affine schemes of modules over gentle algebras. We describe the smooth points of these schemes, and we also analyze their irreducible components in detail. Several of our results generalize formerly known results, e.g. by dropping acyclicity, and by incorporating band modules. A special class of gentle algebras are Jacobian algebras arising from triangulations of unpunctured marked surfaces. For these we obtain a bijection between the set of generically $$\tau $$ τ -reduced decorated irreducible components and the set of laminations of the surface. As an application, we get that the set of bangle functions (defined by Musiker–Schiffler–Williams) in the upper cluster algebra associated with the surface coincides with the set of generic Caldero-Chapoton functions (defined by Geiß–Leclerc–Schröer).

Non-kissing complexes and tau-tilting for gentle algebras

We interpret the support τ \tau -tilting complex of any gentle bound quiver as the non-kissing complex of walks on its blossoming quiver. Particularly relevant examples were previously studied for quivers defined by a subset of the grid or by a dissection of a polygon. We then focus on the case when the non-kissing complex is finite. We show that the graph of increasing flips on its facets is the Hasse diagram of a congruence-uniform lattice. Finally, we study its g \mathbf {g} -vector fan and prove that it is the normal fan of a non-kissing associahedron.

On Extensions for Gentle Algebras

We give a complete description of a basis of the extension spaces between indecomposable string and quasi-simple band modules in the module category of a gentle algebra.