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 這一篇文章是關於 Shokurov 所提3維flip 之存在性證明，文章中將呈現一個清楚的證明排序。而這篇文章主要是從 Corti 的論文架構而起，我希望一個剛剛接觸 Minimal Model Program 的初學者，看了這一篇文章可以了解3維flips之存在性證明的關鍵想法，而且對於往後較深入的研究充滿興趣。
 In this note, I attempt to offer a good-ordered and detailed explanation of Shokurov’s proof for the existence of 3-fold flips. For the most part, it is based on Corti’s paper. I hope that by reading this note, a beginner will catch the key idea of this proof easily and thus be interested in the latest development of Minimal Model Program.
 1 Introduction.......................................................................11.1 Motivation......................................................................11.2 Outline............................................................................21.3 The latest development of MMP.....................................22 Preliminaries......................................................................32.1 Singularities in MMP......................................................32.2 Introduction to MMP......................................................52.3 The existence of flips and reduction to pl flips...............82.4 Function algebra...........................................................122.5 Restricted algebra.........................................................142.6 Sketch of the proof.......................................................153 Shokurov algebra............................................................173.1 b-divisors.....................................................................173.2 Saturated b-divisors......................................................193.3 Sequence of b-divisors.................................................214 How to associate R^0 with a Shokurov algebra?.............234.1 Summary......................................................................234.2 Mobile b-divisors.........................................................234.3 Construction of pbd algebras and Limiting criterion....254.4 Boundedness of pbd algebras......................................304.5 Mobile restriction.........................................................314.6 What is R^S?................................................................325 R^S is finitely generated for surface case........................365.1 Summary.....................................................................365.2 Mobile saturated b-divisers on surfaces.......................385.3 Shokurov algebra and D_X.........................................415.4 D_X isrational.............................................................435.5 Main proof..................................................................44References.........................................................................45
 [CT] Alessio Corti. 3-fold flips after Shokurov, Flips for 3-folds, and 4-folds, Oxford University Press, 2007.[Sh] V.V. Shokurov. Prelimiting flips, Tr. Mat. Inst. Steklova, 240 (Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry): 82- 219,2003, available at http://www.ma.ic.ac.uk/%7Eacorti/flips.html[Ta] Hiromichi Takagi. Pl flips after Shokurov, available at http://www.dpmms.cam.ac.uk/~corti/flips/bookflip3.ps[Fu] Osama Fujino. Special termination and reduction theorem, Flips for 3-folds and 4-folds, Oxford University Press, 2007.[Ha] Robin Hartshorne. Algebraic Geometry. Springer Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52.[Ha1] Robin Hartshorne. Stable Reflexive Sheaves. Math. Ann. 254, 121-176(1980)[KMM] Y. Kawamata, K. Matsuda, K. Matsuki. Introduction to the minimal model problem. In Algebraic geometry, Sendai, 1985, volume 10 of Adv. Stud. Pure Math., pages 283-360. North-Holland, Amsterdam, 1987, available at http://faculty.ms.u-tokyo.ac.jp/~kawamata/index.html[Ka] Y. Kawamata. Termination of log-flips for algebraic 3-folds. Intl. J. Math. 3(1992),653-659, available at http://faculty.ms.u-tokyo.ac.jp/~kawamata/index.html[Ka1] Y. Kawamata. Flops connect minimal models, available at http://arXiv.org ”arXiv:0704.1013v1 [math.AG]”[KM] Kenji Matsuki. Introduction to the Mori Program. Springer-Verlag, New York, 2001.[KM1] Janos Koll’ar and Shigefumi Mori. Birational Geometry of Algebraic Varieties. Cambridge University Press,1998.[Ko] Janos Koll’ar. Flips and Abundance for Algebraic Threefolds. Ast’erisque, 211(1992), 1-258.[BCHM] Caucher Birkar, Paolo Cascini, Christopher D. Hacon, James MCkernan. Existence of minimal models for varieties of log general type, available at http://arXiv.org ”arXiv:0706.1794”[B] Caucher Birkar. Birational Geometry, available at http://arXiv.org ”arXiv:math/0610203”[HM] Christopher D. Hacon and James MCkernan. On the existence of flips, available at http://arXiv.org ”arXiv:math/0507597”[ZS] Zariski, O. and Samuel, P. Commutative Algebra Vol. I. Van Nostrand. [1958-60]
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