四維度之加權映射之研究

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姓名

林皓正(Lin Hao_cheng) 查詢紙本館藏

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數學系

論文名稱

四維度之加權映射之研究
(Weighted Blowups to Cyclic Quotient Terminal Singularity in Dimension 4)

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摘要(中)

這篇論文先介紹森理論中的極小模型理論，接著會看到川又雄二郎先生所計算的結果，他證明在三維度極小模型理論的某種雙有理映射的唯一性。而對於這樣的結果是否能推廣到高維度？事實上，四維度的情形必須要有所限制才能使得該映射唯一。而我在這邊論文裡的計算將會得到一些反例，證明在高維度的情況下存在無限多種的映射。

摘要(英)

In this thesis, we shall introduce the Mori program and minimal model program. Kawamata proved that extremal divisorial contraction X->Y which contracts a divisor to a cyclic quotient terminal singularity is unique for threefold case. However, this result may have trouble in higher dimension. In the end of this thesis, we provide some counterexamples and partial results showing that there may be infinitely many choices of the weighted blowups which contracts a divisor to a cyclic quotient terminal singularity in dimension 4.

關鍵字(中)

★ 加權映射

關鍵字(英)

論文目次

Contents
1 Introduction 1
2 Minimal Model Program 3
2.1 Mori Program . . . . . . . . . . . . . . . . . . . 3
2.2 Minimal Model Program . . . . . . . . . . . . . . 4
3 Weighted blowup 14
3.1 Hayakawa Weighted Blow up . . . . . . . . . . . . 14
3.2 Kawamata Theorem . . . . . . . . . . . . . . . . . 17
4 Main properties 21
4.1 Main Property 1 . . . . . . . . . . . . . . . . . 21
4.2 Main Property 2 . . . . . . . . . . . . . . . . . 24
4.3 Main Property 3 . . . . . . . . . . . . . . . . . 32
4.4 References . . . . . . . . . . . . . . . . . . . . 39

參考文獻

[1 ] Miles Reid, "Young Person′s Guide to Canonical Singularities", Pro-
ceedings of Symposia in Pure Mathematics Volume 46(1987), 345-414.
[2 ] Yujiro Kawamata, "Divisorial contractions to 3-dimensional terminal
quotient singularities", Berlin, 1996.
[3 ] Takayuki Hayakawa, "Blowing Ups of 2-dimensinal Terminal Singulari-
ties", Publ. RIMS, Kyoto Univ. 35(1999), 515-570.
[4 ] Miles Reid,"Graded rings and varieties in weighted projective space",
2002.
[5 ] William, Fulton, "Introduction to Toric Varieties". Princeton University
Press, 1993.
[6 ] Kenji Matsuki, "Introduction to the Mori program". Springer-Verlag
New York, Inc. 2002.
[7 ] Shigefumi Mori, "Threefolds whose canonical bundles are not numeri-
cally effective", Annals of Mathematics, 116(1982), 133-176.
[8 ] Steven Cutkosky, "Elementary Contractions of Gorenstein Threefolds",
Math. Ann. 280, (1988), 521-525.
[9 ] Robin Hartshorne, "Algebraic Geometry", Springer-Verlag, 1997.
[10 ] M.F. Atiyah, I.G. Macdonalb, "Introduction to Commutative Algebra",
Addison-Wesley Publishing Company, 1969.
[11 ] Jheng-Jie Chen, "On Higher Dimensional Singularities", 2010.
[12 ] William, Fulton, "Algebraic curves : an introduction to algebraic ge-
ometry", New York : Benjamin, Inc, 1969.

指導教授

陳正傑

審核日期

2016-1-27

推文