Extensions in Jacobian algebras and cluster categories of marked surfaces

In the context of representation theory of finite dimensional algebras, string algebras are extensively studied and most aspects of their representation theory are well-understood. One exception to the present is that the classification of extensions between indecomposable modules. During this paper we explicitly describe such extensions for a category of string algebras, namely gentle algebras associated to surface triangulations. These algebras arise as Jacobian algebras of unpunctured surfaces. We relate the extension spaces of indecomposable modules to crossings of generalised arcs within the surface and provide explicit bases of the extension spaces for indecomposable modules in most cases. We show that the size of those extension spaces is given in terms of crossing arcs within the surface. Our approach is new and consists of interpreting snake graphs as indecomposable modules. So as to point out that our basis may be a spanning set, we'd like to figure within the associated cluster category where we explicitly calculate the centre terms of extensions and provides bases of their extension spaces.