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: teA}. Then the total space X=UX 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. KMM]). After substantial contributions by several mathematicians (Benveniste, Kawamata, Kollfir, Mori, Reid, Shokurov, Vichweg) 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: seS} is not a flat family of surfaces, and it is of interest to find necessary and sufficient conditions for this to hold. If {J~s: s~S} is a flat family, then K 2-is locally constant on S. One can Xs hope that the converse is also true (at least after a finite and surjective base change). This was proved by Laufer [La, 5.7] under the additional assumption that the Xs have isolated Gorenstein singularities. The main result of chapter two (2.10) generalizes this to the case where the X~ have arbitrary isolated singularities. In the presence of nonisolated singularities the converse is usually false.Chapter three provides a new approach to the study of the deformation space of a quotient singularity. The main result (Theorem 3.9) relates the irreducible components of the deformation space to certain partial resolutions of the singularity. This gives an algorithm to compute the number of components of the deformation space and the dimension of the components in terms of the dual graph of the minimal resolution. It is our belief that this method will lead eventually to the understanding of all small deformations of a quotient singularity, but so far our attempts have been frustrated.The next two chapters contain results pertaining to the problem of compactifying the moduli of surfaces. This involves the study of certain singular surfaces that appear as limits of smooth ones. A good class of such singular surfaces is suggested by the above-mentioned results of three dimensional geometry. The possible singularities of such surfaces...
Perfect quadratic forms give a toroidal compactification of the moduli space
of principally polarized abelian g-folds that is Q-factorial and whose ample
classes are characterized, over any base. In characteristic zero it has
canonical singularities if g is at least 5, and is the canonical model (in the
sense of Mori and Reid) if g is at least 12.Comment: 20 page
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