Singularities of locally acyclic cluster algebras

Abstract: We show that locally acyclic cluster algebras have (at worst) canonical singularities. In fact, we prove that locally acyclic cluster algebras of positive characteristic are strongly F-regular. In addition, we show that upper cluster algebras are always Frobenius split by a canonically defined splitting, and that they have a free canonical module of rank one. We also give examples to show that not all upper cluster algebras are F-regular if the local acyclicity is dropped.