三維 Flip 的存在性之討論

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張詔智(Zhao-Zhi Zhang) 查詢紙本館藏

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論文名稱

三維 Flip 的存在性之討論
(A Discussion of Existence of 3-fold Flips)

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★ A remark on very-ampleness in Toric geometry	★ Special ample divisors in toric 3-folds
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★ 一個解開環面簇的奇異點的有效方法(三維情形)	★ 對角超曲面上奇異點解析的存在性問題
★ Very ampleness of toric 3-folds

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摘要(中)

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。

摘要(英)

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.

關鍵字(中)

★ 存在

關鍵字(英)

★ Flip

論文目次

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

參考文獻

[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]

指導教授

林惠雯(Hui-Wen Lin)

審核日期

2007-11-29

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