Abstract
We study the cluster category of a canonical algebraAAin terms of the hereditary category of coherent sheaves over the corresponding weighted projective lineX\mathbb {X}. As an application we determine the automorphism group of the cluster category and show that the cluster-tilting objects form a cluster structure in the sense of Buan, Iyama, Reiten and Scott. The tilting graph of the sheaf category always coincides with the tilting or exchange graph of the cluster category. We show that this graph is connected if the Euler characteristic ofX\mathbb {X}is non-negative, or equivalently, ifAAis of tame (domestic or tubular) representation type.
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