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On Shokurov’s work “Prelimiting flips”. (English. Russian original) Zbl 1082.14018
Proc. Steklov Inst. Math. 240, 16-36 (2003); translation from Tr. Mat. Inst. Im. V. A. Steklova 240, 21-42 (2003).
Consider a threefold \(X\) and a curve \(C\) such that \(K_X\cdot C<0\). Suppose that there exists a contraction \(\phi\colon X\to Z\) that maps \(C\) to a point \(P\in Z\) and \(\phi\) gives an isomorphism between \(X\setminus C\) and \(Z\setminus \{P\}\). Then the flip of \(\phi\) is another morphism \(\phi^+: X^+\to Z\) and a curve \(C^+\subset X^+\) such that \(K_{X^+}\cdot C^+>0\), \(\phi^+\) contracts \(C^+\) to \(P\) and gives an isomorphism between \(X^+\setminus C^+\) and \(Z\setminus \{P\}\). The definition of a flip in higher dimensions is analogous. Of course the definition of flips does not guarantee their existence. In fact, the question of existence of flips is considered one of the hardest problems in algebraic geometry today. In dimension three it was first proved by S. Mori [J. Am. Math. Soc. 1, No. 1, 117–253 (1988; Zbl 0649.14023)].
V. V. Shokurov proved the existence of (log) flips in dimensions four [in: Birational geometry, Proc. Steklov Inst. Math. 240, 75–213 (2003; Zbl 1082.14019)] and his new approach provides a short, conceptual proof of the existence of threefold flips. The paper under review is a survey of the log minimal model program and Shokurov’s proof of the existence of three-dimensional and four-dimensional log-flips.
For the entire collection see [Zbl 1059.14001].
14E30 Minimal model program (Mori theory, extremal rays)