Schemes of modules over gentle algebras and laminations of surfaces
arXiv:2005.01073 · doi:10.1007/s00029-021-00710-w
Abstract
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.
arXiv comment68 pages. Changes v2: Section 3 is new. We fixed a few typos and added 2 references. v3: Final version with a few improvements suggested by a referee. To appear in Selecta Math. Use PDF-LaTeX for correct display of colored diagrams
References in corpus (14)
- Flat surfaces and stability structures
- Quivers with potentials associated to triangulated surfaces
- Bases for cluster algebras from surfaces
- Locally acyclic cluster algebras
- Dual Teichmuller and lamination spaces
- On the Cluster Category of a Marked Surface
- Generic bases for cluster algebras and the Chamber Ansatz
- A geometric model for the derived category of gentle algebras
- A=U for Locally Acyclic Cluster Algebras
- -tilting finite gentle algebras are representation-finite
- Categorified canonical bases and framed BPS states
- Generic Modules for string algebras
- Moduli spaces of representations of special biserial algebras
- Lattice bijections for string modules, snake graphs and the weak Bruhat order