Schemes of modules over gentle algebras and laminations of surfaces

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