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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.17890 |
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| _version_ | 1866911526509608960 |
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| author | Neville, Scott Simental, José |
| author_facet | Neville, Scott Simental, José |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_17890 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Mysterious points in keys but not trees Neville, Scott Simental, José Combinatorics Algebraic Geometry 13F60, 05C20 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. |
| title | Mysterious points in keys but not trees |
| topic | Combinatorics Algebraic Geometry 13F60, 05C20 |
| url | https://arxiv.org/abs/2603.17890 |