AI 中文总结
针对局部无环簇代数的深轨迹猜想，本文证明了由曲面得到的局部无环曲面型簇代数及有限型簇代数的深点均包含于簇自同构群的稳定子轨迹中，从而验证了这两类簇代数的神秘点猜想。
AI 中文摘要
簇簇代数包含簇环面的并集，不在该并集中的点称为深点，这些点的轨迹称为深轨迹。在arXiv:2402.16970中，针对局部无环簇代数，有人推测了该轨迹的描述，特别指出这应该是簇自同构群的稳定子轨迹。我们针对arXiv:math/0608367中引入的由曲面产生的簇代数，以及arXiv:math/0208229中分类的其余有限型情形，解决了该猜想。特别地，我们证明局部无环曲面型簇簇代数没有不包含在稳定子轨迹中的深点，有限型簇代数也没有不包含在稳定子轨迹中的深点，从而在这些情形下验证了神秘点猜想。
英文摘要
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.
Comments19 pages 10 figures. Comments welcome!
