Poisson geometry and Azumaya loci of cluster algebras

There are two main types of objects in the theory of cluster algebras: the upper cluster algebras U with their Gekhtman–Shapiro–Vainshtein Poisson brackets and their root of unity quantizations Uε. On the Poisson side, we prove that (without any assumptions) the spectrum of every finitely generated upper cluster algebra U with its GSV Poisson structure always has a Zariski open orbit of symplectic leaves and give an explicit description of it. On the quantum side, we describe the fully Azumaya loci of the quantizations Uε under the assumption that Aε=Uε and Uε is a finitely generated algebra. All results allow frozen variables to be either inverted or not.