  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.
