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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

14J30 \(3\)-folds
14B05 Singularities in algebraic geometry