Deformation theory of sandwiched singularities (Q1974874)

A sandwiched singularity is a normal surface singularity which admits a birational map to \(({\mathbb C},0)\). The paper under review is concerned with deformations of sandwiched singularities. Namely, deformations of sandwiched singularities are described via deformations of decorated curves. Resolving a sandwiched singularity, one obtains the total space of a non-minimal embedded resolution of a plane curve singularity. Conversely, given a singular plane curve germ \(C\) and a non-empty zero-dimensional subscheme \(l\) of the normalization of \(C\) (decorated curve), one takes the minimal embedded resolution of \(C\), blows up non-singular points of the strict transform of \(C\), corresponding to the scheme \(l\), then blows down some non-\((-1)\)-rational curves, and obtains a sandwiched singularity \(X(C,l)\). The main theorem states that 1-parametric deformations of \(X(C,l)\) are those which arise from 1-parametric deformations of the decorated curve \((C,l)\) such that \(C\) varies in a \(\delta\)-constant family, \(l\) varies in a flat family on the normalization of the deformation of \(C\). For instance, a deformation of a decorated curve into a curve with ordinary singularities gives rise to a smoothing deformation of the corresponding sandwiched singularity. In this case, a lot of topological information on the Milnor fibre of a sandwiched singularity can be extracted from geometry of the deformation of the decorated curve. General results are illustrated by many interesting examples.