an:05310666
Zbl 1145.14014
Kawamata, Yujiro
Flops connect minimal models
EN
Publ. Res. Inst. Math. Sci. 44, No. 2, 419-423 (2008).
00222663
2008
j
14E30 14E05 14J32
minimal models; flops
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 \textit{C. Birkar, P. Cascini, J. McKernan} and the reviewer [Existence of minimal models for varieties of log general type, Preprint, \url{arXiv:math/0610203}] and on a result concerning the boundedness of the length of extremal rays.
Christopher Hacon (Salt Lake City)