Weighted Projective Lines and Rational Surface Singularities

In this paper we study rational surface singularities R with star shaped dual graphs, and under very mild assumptions on the self-intersection numbers we give an explicit description of all their special Cohen-Macaulay modules. We do this by realising R as a certain Z-graded Veronese subring S^x of the homogeneous coordinate ring S of the Geigle-Lenzing weighted projective line X, and we realise the special CM modules as explicitly described summands of the canonical tilting bundle on X. We then give a second proof that these are special CM modules by comparing qgr S^x and coh X, and we also give a necessary and sufficient combinatorial criterion for these to be equivalent categories. In turn, we show that qgr S^x is equivalent to qgr of the reconstruction algebra, and that the degree zero piece of the reconstruction algebra coincides with Ringel's canonical algebra. This implies that the reconstruction algebra contains the canonical algebra, and furthermore its qgr category is derived equivalent to the canonical algebra, thus linking the reconstruction algebra of rational surface singularities to the canonical algebra of representation theory. Comment: Final version

Invariants of open books of links of surface singularities

If M is the link of a complex normal surface singularity, then it carries a canonical contact structure ξcan, which can be identified from the topology of the 3-manifold M. We assume that M is a rational homology sphere. We compute the support genus, the binding number and the norm associated with the open books which support ζcan, provided that we restrict ourselves to the case of (analytic) Milnor open books. In order to do this, we determine monotonity properties of the genus and the Milnor number of all Milnor fibrations in terms of the Lipman cone.We generalize results of [3] valid for links of rational surface singularities, and we answer some questions of Etnyre and Ozbagci [7, section 8] regarding the above invariants.