Scattering Fans

Scattering diagrams arose in the context of mirror symmetry, Donaldson-Thomas theory, and integrable systems. We show that a consistent scattering diagram with minimal support cuts the ambient space into a complete fan. A special class of scattering diagrams, the cluster scattering diagrams, is closely related to cluster algebras. We show that the cluster scattering fan associated to an exchange matrix B refines the mutation fan for B (a complete fan that encodes the geometry of mutations of B). We conjecture that, when B is n x n for n > 2, these two fans coincide if and only if B is of finite mutation type.