  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-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.
