Deep Points of Cluster Varieties

Abstract

A cluster algebra determines an associated cluster variety, and each cluster determines a cluster torus in that variety. The cluster tori sometimes cover the entire cluster variety, but not always; we refer to any points not contained in any cluster torus as deep points. Any geometrically interesting points in the variety (e.g. singularities) must be deep, and many pathologies of the cluster algebra can be localized to those deep points. We describe the deep points of cluster algebras of unpunctured polygons, unpunctured marked surfaces, punctured polygons, and punctured surfaces with at least two boundary marked points. As a consequence, we classify the deep points of cluster algebras of types A_n and D_n. We also classify the deep points of the Markov cluster algebra — the cluster algebra of the once-punctured torus — and its upper cluster algebra. We then examine the deep points of skew-symmetrizable cluster algebras that arise from a folding of an acyclic quiver by a finite group of automorphisms. In particular, we study the deep points of cluster algebras of types B_n, C_n, and F_4 via foldings of quivers of types A_{2n-1}, D_{n+1} and E_6, respectively.