Locally Acyclic Surface Type and Finite Type Cluster Algebras Have No Mysterious Points

Abstract

Cluster varieties contain the union of cluster tori and the points not in this union are called $\textit{deep points}$, and the locus of these points is called the $\textit{deep locus}$. In arXiv:2402.16970, a description of this locus is conjectured for locally acyclic cluster algebras, in particular, stating that this should be the stabilizer locus of the cluster automorphism group. We resolve this conjecture for the case of cluster algebras arising from surfaces, introduced in arXiv:math/0608367, and the remaining finite type cases as categorized in arXiv:math/0208229. In particular, we show locally acyclic surface type cluster varieties have no deep points not contained in the stabilizer locus, and finite type cluster algebras also have no deep points not contained in the stabilizer locus, validating the mysterious points conjecture in these cases.