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.......................................................................1
1.1 Motivation......................................................................1
1.2 Outline............................................................................2
1.3 The latest development of MMP.....................................2
2 Preliminaries......................................................................3
2.1 Singularities in MMP......................................................3
2.2 Introduction to MMP......................................................5
2.3 The existence of flips and reduction to pl flips...............8
2.4 Function algebra...........................................................12
2.5 Restricted algebra.........................................................14
2.6 Sketch of the proof.......................................................15
3 Shokurov algebra............................................................17
3.1 b-divisors.....................................................................17
3.2 Saturated b-divisors......................................................19
3.3 Sequence of b-divisors.................................................21
4 How to associate R^0 with a Shokurov algebra?.............23
4.1 Summary......................................................................23
4.2 Mobile b-divisors.........................................................23
4.3 Construction of pbd algebras and Limiting criterion....25
4.4 Boundedness of pbd algebras......................................30
4.5 Mobile restriction.........................................................31
4.6 What is R^S?................................................................32
5 R^S is finitely generated for surface case........................36
5.1 Summary.....................................................................36
5.2 Mobile saturated b-divisers on surfaces.......................38
5.3 Shokurov algebra and D_X.........................................41
5.4 D_X isrational.............................................................43
5.5 Main proof..................................................................44
References.........................................................................45

[CT] Alessio Corti. 3-fold flips after Shokurov, Flips for 3-folds, and 4-folds, Oxford University Press, 2007.
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[Ta] Hiromichi Takagi. Pl flips after Shokurov, available at http://www.dpmms.cam.ac.uk/~corti/flips/bookflip3.ps
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