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Smooth models for elliptic threefolds. (English) Zbl 0583.14014
Birational geometry of degenerations, Summer Algebraic Geometry Semin., Harvard Univ. 1981, Prog. Math. 29, 85-133 (1983).
[For the entire collection see Zbl 0493.00005.]
The author considers a threefold $$X_ 0$$ endowed with a proper map $$f_ 0: X_ 0\to S_ 0$$ onto a surface $$S_ 0$$, whose generic fibre is a smooth elliptic curve. Under the assumption that $$f_ 0$$ admits a section he constructs a smooth model $$f: X\to S$$ of $$X_ 0$$ having the following properties: (1) X and S are smooth; (2) all fibres of f are one dimensional; (3) X contains no generically contractible surface whose contractible fibres lie in the fibres of f; (4) the discriminant locus $$D\subset S$$ of f is a curve with at worst ordinary double points as singularities; (5) at a smooth point $$p\in D$$, $$f^{-1}(p)$$ is a singular elliptic curve as in Kodaira’s list and his type is locally constant near p; (6) at $$p\in Sing(D)$$, $$f^{-1}(p)$$ is determined by the singular fibre types of the two branches of D at p. - Let $$S_ 1$$ be the open subset of $$S_ 0$$ of smooth points p such that $$f^{-1}(p)$$ is smooth and let $$X_ 1=f^{-1}(S_ 1)$$; the existence of a section for the fibration $$f_ 1=f| X_ 1: X_ 1\to S_ 1$$ allows the author to write $$f_ 1$$ in the Weierstraß form and by compactifying $$S_ 1$$ to a smooth surface S he gets a new threefold $$\pi$$ : $$W\to S$$; W extends the smooth locus of the original elliptic threefold and near any point $$p\in S$$, it can be written as $$y^ 2=x^ 3+ax+b$$, where a, b are regular functions on S near p. In general W is singular; the most part of the paper is concerned with the description of the explicit desingularization of W. W is a double cover of a $${\mathbb{P}}^ 1$$-bundle over S whose branch locus consists of the section $$x=\infty$$ and of the 3-section defined by $$x^ 3+ax+b=0$$. So the desingularization of W is obtained by performing embedded resolution of the branch locus in the $${\mathbb{P}}^ 1$$-bundle and by taking the double cover. A hard point is the computation of $$f^{-1}(p)$$, $$p\in Sing(D)$$, given the two singular fibre types near p.
Reviewer: A.Lanteri

##### MSC:
 14J30 $$3$$-folds 14B05 Singularities in algebraic geometry