Threefolds and deformations of surface singularities

TLDR: In this paper, the authors studied the study of surface singularities using recent advances in 3D geometry and proved the existence of a minimal resolution of singularities for a given set of surfaces.

Abstract: The central theme of this article is the study of deformations of surface singularities using recent advances in three dimensional geometry. The basic idea is the following. Let X 0 be a surface singularity and consider a one parameter deformation {Xo: t e A } . Then the total space X = U X t is a three dimensional object. One can attempt to use the geometry of X to get information about the surface X~. In general X is very singular and so one can try to study it via a suitable resolution of singularities f : X ' -~ X. The existence of a resolution was established by Zariski; the problem is that there are too many of them, none particularly simple. Mori and Reid discovered that the best one can hope for is a partial resolution f : X ' ~ X where X' possesses certain mild singularities but otherwise is a good analog of the minimal resolution of surface singularities. The search for such a resolution is known as Mori 's program (see e.g. [-Ko3, KMM]). After substantial contributions by several mathematicians (Benveniste, Kawamata, Kollfir, Mori, Reid, Shokurov, Vichweg) this was recently completed by Mori [Mo 3]. A special case, which is nonetheless sufficient for the applications presented here, was settled by several persons. A proof was first announced by Tsunoda [TsM], later followed by Shokurov [Sh], Mori [Mo2] and Kawamata [Kaw2]. A precise formulation of the result we need will be provided at the end of the introduction. In certain situations X0 will impose very strong restrictions on X ' and one can use this to obtain information about X and X~ for t 40 . The first application is in chapter two. Teissier [Tel posed the following problem. Let {X~ : s~S} be a flat family of surfaces parameterized by the connected space S. Let X s be the minimal resolution of X~. In general {Xs: s e S } is not a flat family of surfaces, and it is of interest to find necessary and sufficient conditions for this to hold.