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Flops connect minimal models. (English) Zbl 1145.14014
Let \((X,B)\) and \((X',B')\) be two Kawamata log terminal pairs and \(p:X\to Z\) and \(q:X'\to Z\) be two birational morphisms that contract no divisors. The rational map \((q^{-1})\circ p:X\dasharrow X'\) is a flop if \(\rho (X/Z)=\rho (X'/Z)=1\), \(B'=(q^{-1}\circ p)_*B\) and both \(K_X+B\) and \(K_{X'}+B'\) are numerically equivalent to the pull back of a divisor from \(Z\). The minimal model program predicts that if \((X,B)\) and \((X',B')\) are two minimal models, then \(q^{-1}\circ p\) is given by a finite sequence of flops. In the paper under review, the author shows that this is indeed the case. More precisely, he shows that if \((X,B)\) and \((X',B')\) are two projective \(\mathbb Q\)-factorial terminal pairs where \(K_X+B\) and \(K_{X'}+B'\) are nef and if \(\alpha : X \dasharrow X'\) is a birational map such that \(\alpha _* B=B'\), then \(\alpha\) may be decomposed in to a sequence of flops \(\alpha = \alpha _t\circ \ldots \circ \alpha _1\). The proof is based on a result of C. Birkar, P. Cascini, J. McKernan and the reviewer [Existence of minimal models for varieties of log general type, Preprint, arXiv:math/0610203] and on a result concerning the boundedness of the length of extremal rays.

MSC:
14E30 Minimal model program (Mori theory, extremal rays)
14E05 Rational and birational maps
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
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[1] C. Birkar, P. Cascini, C. D. Hacon, J. McKernan, Existence of minimal models for varieties of log general type, math.AG/0610203. · Zbl 1210.14019
[2] Y. Kawamata, Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces, Ann. of Math. (2) 127 (1988), no. 1, 93-163. · Zbl 0651.14005 · doi:10.2307/1971417
[3] , On the length of an extremal rational curve, Invent. Math. 105 (1991), no. 3, 609-611. · Zbl 0751.14007 · doi:10.1007/BF01232281 · eudml:143926
[4] , On the cone of divisors of Calabi-Yau fiber spaces, Internat. J. Math. 8 (1997), no. 5, 665-687. · Zbl 0931.14022 · doi:10.1142/S0129167X97000354 · arxiv:alg-geom/9701006
[5] J. Kollár, Flops, Nagoya Math. J. 113 (1989), 15-36.
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