Topological Models for Categories Associated with the Weighted Projective Lines of Type $(2,2,2,2)$

Abstract

We construct a geometric model for the derived category of the weighted projective line ${\mathbb{CP}}_{\omega }^1$ 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 $\mathrm{Hom}$-spaces are computed by oriented intersection numbers. As applications, we provide a geometric realization of the indecomposable rigid objects in the cluster category $\mathcal{C}({\mathbb{CP}}_{\omega }^1)$ via tagged arcs on the four-punctured sphere, with $\mathrm{Hom}$-space dimensions given by tagged intersection numbers. When the weighted points of ${\mathbb{CP}}_{\omega }^1$ 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.

47 pages, 9 figures