与型为$(2,2,2,2)$的加权射影直线相关范畴的拓扑模型
Topological Models for Categories Associated with the Weighted Projective Lines of Type $(2,2,2,2)$
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中文总结 AI 辅助
本文为型为$(2,2,2,2)$的加权射影直线的导出范畴构造了几何模型，给出不可分解刚性对象与弧的双射，实现了cluster范畴中不可分解刚性对象的几何构造，且对应关系与相关群作用相容。
中文摘要 AI 辅助
我们利用带有四个二元组的分次球面，为型为$(2,2,2,2)$的加权射影直线$\boldsymbol{\text{CP}}^1_{\boldsymbol{\text{ω}}}$的导出范畴构造了一个几何模型。更确切地说，我们给出了不可分解刚性对象集合与特定弧集合之间的双射，使得$\boldsymbol{\text{Hom}}$空间的维数可通过定向相交数计算。作为应用，我们通过四孔球面上的标记弧，给出了cluster范畴$\boldsymbol{\text{C}}(\boldsymbol{\text{CP}}^1_{\boldsymbol{\text{ω}}})$中不可分解刚性对象的几何实现，其$\boldsymbol{\text{Hom}}$空间维数由标记相交数给出。当$\boldsymbol{\text{CP}}^1_{\boldsymbol{\text{ω}}}$的加权点为$(0,1,\boldsymbol{\text{∞}},\frac{1}{2})$时，这两种对应关系均与自同构群及对应的映射类群的自然作用相容。
英文摘要
We construct a geometric model for the derived category of the weighted projective line $\mathbb{CP}^1_ω$ of type $(2,2,2,2)$ using a graded sphere with four binaries. More precisely, we give a bijection between the set of indecomposable rigid objects and the set of certain arcs, such that the dimensions of $\operatorname{Hom}$-spaces are computed by oriented intersection numbers. As applications, we provide a geometric realization of the indecomposable rigid objects in the cluster category $\mathcal{C}(\mathbb{CP}^1_ω)$ via tagged arcs on the four-punctured sphere, with $\operatorname{Hom}$-space dimensions given by tagged intersection numbers. When the weighted points of $\mathbb{CP}^1_ω$ are $(0,1,\infty,\frac{1}{2})$, both correspondences are compatible with the natural actions of the automorphism groups and the corresponding mapping class groups.
