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Threefolds and deformations of surface singularities. (English) Zbl 0642.14008

The authors study the deformation of the surface singularities, using the minimal model theory by Reid, Mori etc.
In § 2, a result of Laufer is generalized: Let \(f: X\to Y\) be a flat family of projective surfaces with isolated singularities only such that Y is semi-normal. Then \((i)\quad \bar K^ 2_ Y\) is lower semi- continuous; \((ii)\quad \bar K^ 2_ Y\) is locally constant if and only if f admits a simultaneous Du Val resolution; \((iii)\quad Ifp: Y\to X\) is a section and Y is connected, then f admits a weak simultaneous resolution along p if and only if the germs of \(X_ y\) along p are pairwise homeomorphic.
In § 3, a deformation of the quotient singularity is considered. The normal surface singularity is called of class T if it is a quotient singularity and it admits a \({\mathbb{Q}}\)-Gorenstein one-parameter smoothing. A P-resolution of \(X_ 0\) is a partial resolution \(g: Z_ 0\to X_ 0\) such that \(Z_ 0\) has only singularities of class T. The authors show a one-one correspondence between the components of \(Def(X_ 0)\) (the deformation space) and the P-resolutions of \(X_ 0\). In § 4, they classify the semi-canonical (or terminal) singularities without the normality assumption. This is a generalization of a result of Kawamata. - In § 5, the moduli space of surface of general type is considered. The authors show that appropriate singularities to permit on the surfaces at the boundaries of the moduli space are semi-log-canonical. They also make precise the moduli problem. - In § 7, they show that any small deformation \((X_ t,x_ t)\) of a cyclic quotient singularity is also a cyclic quotient singularity which was conjectured by Riemenschneider.
Reviewer: M.Oka

MSC:

14J17 Singularities of surfaces or higher-dimensional varieties
14B07 Deformations of singularities
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
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