Mysterious points in keys but not trees

TL;DR

This paper investigates the deep locus of cluster varieties, revealing the existence of mysterious points in certain cases, which challenges previous conjectures and enhances understanding of cluster algebra structures.

Contribution

It characterizes the deep locus for cluster varieties with tree seeds and identifies the presence of mysterious points in various acyclic quivers, refuting a prior conjecture.

Findings

Deep locus characterized by nontrivial stabilizers in certain cluster varieties.

Mysterious points exist in many acyclic quivers, including keys.

Refutes Conjecture 1.1 in specific cases.

Abstract

The deep locus of a cluster variety is defined to be the set of its points that do not belong to any cluster torus. We show that, if the cluster variety has a seed whose mutable part is a tree without multiple edges, then the deep locus can be characterized as the set of points whose stabilizer under a certain group action is nontrivial. Deep points without a stabilizer are called mysterious. We establish that many other classes of acyclic quivers (including keys) often have mysterious points. This refutes Conjecture 1.1 of arXiv:2402.16970, but establishes it in many important cases.