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Decomposition of a birational map of three-dimensional varieties outside codimension 2. (English. Russian original) Zbl 0544.14024

Math. USSR, Izv. 21, 187-200 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, 881-895 (1982).
The following important theorems are proved: 1. If f:\(X\to Y\) is a birational transformation of non-singular three-dimensional projective manifolds over an algebraically closed field k, \(char k=0,\) then there exist morphisms \(\eta\) and \(\tau\), being compositions of \(\sigma\)- processes with \(\tilde X\to^{\tilde f}\tilde Y\to^{\tau}Y\) and \(\tilde X\to^{\eta}X\to^{f}Y\) where \(\tau \circ \tilde f=f\circ \eta,\) and such that birational transformations \(\tilde f\) and \(\tilde f{}^{-1}\) are isomorphisms outside co-dimension 2 (do not contract divisors) and sets of their indeterminacy consist of unions of non- singular rational curves, intersecting each other transversely. Here \(\sigma\)-processes are monoidal transformations with non-singular centres; transversely intersecting curves are curves \(C_ 1\subset X\) and \(C_ 2\subset X\) such that in any point \(x\in C_ 1\cap C_ 2\) the dimension of intersection of tangent spaces \(\dim(T_ x(C_ 1)\cap T_ x(C_ 2))=0.\)- 2. Let f:\(X\to Y\) be an eigen birational morphism of non- singular algebraic three-dimensional manifolds over an algebraically closed field k, \(char k=0,\) and let Y be quasi-projective. Then there a exist non-singular manifold Z and a morphism g:\(Z\to Y\) such that: \((a)g^{-1}\circ g\) is a composition of \(\sigma\)-processes and inverse transformations for them, (b) \(g^{-1}\) is undefined only in isolated points. B. Grauder informed the author that he had also obtained a result which is analogous to theorem 2.
Reviewer: V.F.Ignatenko

MSC:

14J30 \(3\)-folds
14E05 Rational and birational maps
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