An Effective Bound For Sarkisov Program In dimension 2

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蔡銘家(CAI MING JIA) 查詢紙本館藏

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論文名稱

(An Effective Bound For Sarkisov Program In dimension 2)

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摘要(中)

這篇碩士論文在學習代數曲面上的Sarkisov Program：即固定兩個Mori fiber spaces間的雙有理映射可拆解成有限多Sarkisov links。從論述過程中，我們可估計所需的Sarkisov links個數上界。針對這個目標，文中介紹解決此問題所需的背景知識(R.Hartshorne撰寫的[Algebraic Geometry]第一到三章和第五章以及K.Matsuki撰寫的[Introduction to the Mori Program]第一章)；此外，我們介紹2維度的Minimal Model Program兩種產物:Minimal model和Mori fiber spaces及其性質。

摘要(英)

The motivation of this thesis is to study Sarkisov Program in dimension 2 : any birational map between two Mori fiber spaces can be decomposed into finitely many Sarkisov links. According to the study, we are able to estimate a specific upper bound for the number of Sarkisov links in the program. For this purpose, we give some basic terminologies ([R.Hartshorne, “Algebraic Geometry”] from chapter 1 to chapter 3 and chapter 5, [K.Matsuki, “Introduction to the Mori Program”]chapter 1). In addition, we introduce the minimal model program and some properties of its two outcomes: minimal models and Mori fiber spaces.

關鍵字(中)

★ 代數曲面

關鍵字(英)

★ minimal model
★ mori fiber space
★ cone theorem
★ Sarkisov program

論文目次

1 Introduction 1
2 Fundamental Knowledge of Algebraic Geometry 4
2.1 Varieties and Rational Maps . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Sheaves and Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Sheaves of modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Cone Theorem of dimension 2 24
4 Basic Properties of Mori Fiber Spaces and Minimal Models in Dimen
sion 2 27
5 Birational Relation Among Surfaces 30
6 Main Theorem 46
7 Cone Theorem in higher dimension 54
Reference 61

參考文獻

[1] A. Corti, Factoring birational maps of 3-folds after Sarkisov, Journal al
gebraic Geometry 4 (1995), 223-254.
[2] C. Hacon and J. MacKernan The Sarkisov Program, Journal algebraic
Geometry 22 (2013), 389-405.
[3] R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics
52, Springer-Verlag, New York, Heidelberg, Berlin, 1977.
[4] K. Matsuki, Introduction to the Mori Program
[5] J. Koll ´ ar and S. Mori ”Birational geometry of algebraic varieties.”
With the collaboration of C. H. Clemens and A. Corti. Translated
from the 1998 Japanese original. Cambridge Tracts in Mathematics, 134.
Cambridge University Press, Cambridge, 1998.
[6] Y. Kawamata, K. Matsuda and K. Matsuki, Introduction to the minimal
model problem, Algebr. Geom. (Sendai 1985), Adv. Stud. Pure Math.
10, North-Holland, Amsterdam (1987), 283– 60.
[7] W. Fulton, Algebraic Curves, New York, Benjamin, Inc, 1969.

指導教授

陳正傑

審核日期

2018-5-23

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