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We classify the thick subcategories of discrete derived categories. To do this we introduce certain generating sets called arc-collections which correspond to configurations of non-crossing arcs on a geometric model. We show that every thick subcategory is generated by an arc-collection, each thick subcategory is determined by the topology of the corresponding configuration, and we describe a version of mutation which acts transitively on the set of arc-collections generating a given thick subcategory., Comment: 42 pages, 10 figures

We complete the derived equivalence classification of the gentle two-cycle algebras initiated in earlier papers by Avella-Alaminos and Bobinski-Malicki.

A cluster tilted algebra is known to be gentle if and only if it is cluster tilted of Dynkin type $\bbA$ or Euclidean type $\tilde{\bbA}$. We classify all finite dimensional algebras which are derived equivalent to gentle cluster tilted algebras.

In association with a finite dimensional algebra A of global dimension two, we consider the endomorphism algebra of A, viewed as an object in the triangulated hull of the orbit category of the bounded derived category, in the sense of Amiot. We characterize the algebras A of global dimension two such that its endomorphism algebra is isomorphic to a cluster-tilted algebra with a cyclically oriented quiver.Furthermore, in the case that the cluster tilted algebra with a cyclically oriented quiver is of Dynkin or extended Dynkin type then A is derived equivalent to a hereditary algebra of the same type., Comment: 14 pages, 8 figures

We study the cluster category of a canonical algebra A in terms of the hereditary category of coherent sheaves over the corresponding weighted projective line 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-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 of X is non-negative, or equivalently, if A is of tame (domestic or tubular) representation type., Comment: 16 pages. A few minor changes, one reference added. To appear in Trans. Amer. Math. Soc