We investigate the triangle singularity f = x a + y b + z c , or S = k[x, y, z]/(f ), attached to the weighted projective line X given by a weight triple (a, b, c). We investigate the stable category vect-X of vector bundles on X obtained from the vector bundles by factoring out all the line bundles. This category is triangulated and has Serre duality. It is, moreover, naturally equivalent to the stable category of graded maximal Cohen-Macaulay modules over S (or matrix factorizations of f ), and then by results of Buchweitz and Orlov to the singularity category of f .We show that vect-X is fractional Calabi-Yau whose CY-dimension is a function of the Euler characteristic of X. We show the existence of a tilting object which has the shape of an (a − 1) × (b − 1) × (c − 1)-cuboid. Particular attention is given to the weight types (2, a, b) yielding an explanation of Happel-Seidel symmetry for a class of important Nakayama algebras. In particular, the weight sequence (2, 3, p) corresponds to an ADE-chain, the Enchain, extrapolating the exceptional Dynkin cases E 6 , E 7 and E 8 to a whole sequence of triangulated categories.

Abstract. 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 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 of X is non-negative, or equivalently, if A is of tame (domestic or tubular) representation type.

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Abstract. We show a surprising link between singularity theory and the invariant subspace problem of nilpotent operators as recently studied by C. M. Ringel and M. Schmidmeier, a problem with a longstanding history going back to G. Birkhoff. The link is established via weighted projective lines and (stable) categories of vector bundles on those.The setup yields a new approach to attack the subspace problem. In particular, we deduce the main results of Ringel and Schmidmeier for nilpotency degree p from properties of the category of vector bundles on the weighted projective line of weight type (2, 3, p), obtained by Serre construction from the triangle singularity x 2 + y 3 + z p . For p = 6 the Ringel-Schmidmeier classification is thus covered by the classification of vector bundles for tubular type (2, 3, 6), and then is closely related to Atiyah's classification of vector bundles on a smooth elliptic curve.Returning to the general case, we establish that the stable categories associated to vector bundles or invariant subspaces of nilpotent operators may be naturally identified as triangulated categories. They satisfy Serre duality and also have tilting objects whose endomorphism rings play a role in singularity theory. In fact, we thus obtain a whole sequence of triangulated (fractional) Calabi-Yau categories, indexed by p, which naturally form an ADE-chain.