A note on the toric duality between the cyclic quotient surface singularities $A_{n,q}$ and $A_{n,n - q}$

Abstract

In my lecture at the Franco-Japanese Symposium on Singularities I gave an introduction to the work of Martin Hamm [3] 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. [8]), I concentrate in the present note on some other aspect of [3]: the toric duality of the total spaces of the deformations over the monodromy coverings of the Artin components for the singularities $A_{n,q}$ and $A_{n,n - q}$ 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 $q = n - 1$ in which the interplay between algebra and geometry of the underlying polyhedral cones is rather obvious, especially when bringing also the “complementarity” of $A_{n,q}$ and $A_{n,n - q}$ into the game. We then treat the dual case $q = 1$ 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).