On equisingular deformations of cone-like surface singularities

In this note we study the rigidity-problem in the equisingular deformation theory for normal surface singularities whose exceptional sets of their minimal resolutions are smooth. We show that they admit non-trivial equisingular deformations if they are non-rational and if their analytic structures are not too different from those of cones. Latter condition is e.g. automatically satisfied if the absolute value of the selfintersection number of the exceptional set A is not less than the genus of A.