Topological model for derived category associated to sphere with four binaries

2026/07/31 by Jianmin Chen, Li Fan, Yu Qiu +1
Mathematics · #math.RT #msc:18G80 #msc:16G20 #msc:57K20

paper · pdf

47 pages, 9 figures

arxiv created 2026/07/31 · arxiv updated 2026/08/04

Abstract

We construct a geometric model for the derived category of the weighted projective line \mathbbCP¹_ω of type (2,2,2,2), given by a graded sphere with four binaries. Precisely, we give a bijection between the set of indecomposable rigid objects and the set of certain arcs, such that the dimensions of Hom-spaces are computed by oriented intersection numbers. As applications, we provide a geometric realization of the indecomposable rigid objects in the cluster category C(\mathbbCP¹_ω) via tagged arcs on the sphere with four punctures, where Hom-space dimensions are expressed in terms of tagged intersection numbers. When the weighted points of \mathbbCP¹_ω are (0,1,∞,(1)/(2)), both the correspondences are compatible with the natural actions of the automorphism groups and mapping class groups.

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