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Algebraic varieties of dimension three whose hyperplane sections are Enriques surfaces. (English) Zbl 0612.14041
Three-dimensional varieties whose general hyperplane sections are Enriques surfaces are studied. Such 3-folds have been treated by G. Fano and the purpose of the authors is to analyze the results of Fano from a modern point of view. - Let \(W\subset {\mathbb{P}}^ N\) be a 3-fold satisfying the conditions: (i) W is projectively normal, (ii) the general hyperplane section of W is an Enriques surface, (iii) W is not a cone. Then W necessarily has singular points \(P_ 1,...,P_ n\). Under some more ”general case” conditions (for example, (iv) if \(\pi: W\to W\) is the blowing up of \(P_ 1,...,P_ n\), then \(\pi^{-1}(P_ i)\) is smooth for every i, (v) the singularities \(P_ 1,...,P_ n\) are ”similar”), the following theorem is proved:
W has eight singular points \(P_ 1,...,P_ 8\), each of which has the cone over a Veronese surface as its tangent cone. W contains a linear system \(| \phi |\) of Weil divisors whose general member is a K3 surface and the image of the rational map defined by \(| \phi |\) is a Fano variety of degree \(2p-6\) (where p is the sectional genus of W and \(\deg (W)=2p-2).\)
The authors give examples of such 3-folds for the cases \(p=4, 6, 7, 9\) and 13.
Reviewer: K.Watanabe

MSC:
14J30 \(3\)-folds
14J25 Special surfaces
14J17 Singularities of surfaces or higher-dimensional varieties
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