A note on the toric duality between the cyclic quotient surface singularities and
Oswald RiemenschneiderUniversität Hamburg, Germany
A subscription is required to access this book chapter.
In my lecture at the Franco-Japanese Symposium on Singularities I gave an introduction to the work of Martin Hamm  concerning the explicit construction of the versal deformation of cyclic surface singularities. Since that part of his dissertation is already documented in a survey article (cf. ), I concentrate in the present note on some other aspect of : the toric duality of the total spaces of the deformations over the monodromy coverings of the Artin components for the singularities and which themselves are toric duals of each other. Our exhibition is based – as in Hamm’s dissertation – on the algebraic aspects, i.e., the algebras and their generators of these total spaces. We prove Hamm’s remarkable duality result in this note first in detail for the hypersurface case in which the interplay between algebra and geometry of the underlying polyhedral cones is rather obvious, especially when bringing also the “complementarity” of and into the game. We then treat the dual case of cones over the rational normal curves once more in order to develop the necessary ideas for transforming the generators in such a way that it becomes transparent how to compute the dual, even in the general situation (which we explain in the last section by an example).