On the Milnor Fibers of Sandwiched Singularities

The sandwiched singularities are those rational surface singularities that dominate birationally smooth surface singularities. de Jong and van Straten showed that one can reduce the study of the deformations of a sandwiched singularity to the study of deformations of a one-dimensional object, a so-called decorated plane curve singularity. In particular, the Milnor fibers corresponding to their various smoothing components may be reconstructed up to diffeomorphisms from those deformations of associated decorated curves that have only ordinary singularities. Part of the topology of such a deformation is encoded in the incidence matrix between the irreducible components of the deformed curve and the points that decorate it, well defined up to permutations of columns. Extending a previous theorem of ours, which treated the case of cyclic quotient singularities, we show that the Milnor fibers that correspond to deformations whose incidence matrices are different up to permutations of columns are not diffeomorphic in a strong sense. This gives a lower bound on the number of Stein fillings of the contact boundary of a sandwiched singularity.