Singularities of surfaces and threefolds
arxiv: Mathematics::Algebraic Geometry
This thesis has two parts, summarized below. The links between them are discussed at the end of this introduction.\ud \ud Part 1 is concerned with the problem of giving necessary and sufficient conditions for a family of surfaces to have a simultaneous resolution; this property can be regarded as a very weak form of equissingularity (cf. [Te]). I conjecture that, roughly speaking, the plurigenera Pn of a family of singular surfaces of general type are upper semi-continuous and that simultaneous resolution is possible if and only if Pn is locally constant for some n>=2 (equivalently, fo all n>=2). Two cases of this conjecture are proved, under different hypotheses on the special fibre. The techniques used are the use of adjunction ideals, suggested to me by Reid, and the results of Brieskorn, Tyurina and others on deformations of Du Val singularities (also known as rational double points, ...). A very similar approach was used by Lipman [Li] for the study of deformations of arbitrary rational singularities.\ud \ud Part 2 is concerned with canonical singularities, as defined by Reid [R3]. We first prove that in dimensions <=4 they are Cohen-Macaulay, and then deduce a corollary on the invariance of plurigenera in some special circumstances; this answers, in part, questions asked me by Reid. Since these results were proved, Elkik and Gabber have shown that canonical singularities are Cohen-Macaulay in all dimensions. We then consider some specific classes of singularities, and prove that they are canonical. The idea of using the techniques and results of Kulikov in this situation was suggested to me by Dave Morrison, and I subsequently learnt that Theorem 5 was already known to him and others, including Pinkham and Wahl. The point of this sections is twofold; firstly it gives an analysis of what are the simplest canonical singularities, and secondly it shows quite explicitly that the contractibility of a given configuration of surfaces in a 3-fold is a much more delicate question than in the case of curves lying on a surface. The problem of contractibility underlies Chapter 1 as well; a sufficiently strong result would kill certain cohomology groups that are the obstruction to proving the conjecture.