International Mathematics Research Notices Advance Access published online on October 30, 2009
International Mathematics Research Notices, doi:10.1093/imrn/rnp167
On the Milnor Fibers of Sandwiched Singularities
1 Renyi Institute of Mathematics, POB 127, H-1364 Budapest, Hungary
2 Université Paris 7 Denis Diderot, Institut de Mathématiques—UMR CNRS 7586, équipe "Géométrie et dynamique," Site Chevaleret, Case 7012, 75205 Paris Cedex 13, France
Correspondence: Correspondence to be sent to: ppopescu{at}math.jussieu.fr
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.