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Simultaneous resolution of threefold double points. (English) Zbl 0576.14013

This paper is concerned with the deformation theory of isolated hypersurface singularities of dimension three with a small resolution, i.e. a resolution of singularities whose exceptional set has dimension 1. If X denotes the germ of such a singularity and \(\tilde X\) a small resolution of it, there is an induced morphism of deformation functors Def \(\tilde X\to Def X\). It is shown that this morphism is unramified and its image is studied. Next the global problem is considered. Let X denote either a generalized Fano variety or a (singular) threefold with trivial dualizing sheaf. Precise results are obtained in either case when all singularities are ordinary double points. As a corollary of the result on Fano threefolds, the maximum number of nodes on a quartic threefold in \({\mathbb{P}}^ 4\) is shown to be 45. The results on varieties with trivial dualizing sheaf are used to construct examples of non-Kähler threefolds with trivial canonical bundle. Finally, there are some applications to the behavior of rational curves of low degree on quintic threefolds.

MSC:

14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14J30 \(3\)-folds
14B07 Deformations of singularities
14D15 Formal methods and deformations in algebraic geometry
14J25 Special surfaces
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J17 Singularities of surfaces or higher-dimensional varieties
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References:

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