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研究生:林金毅
研究生(外文):Chin-Yi Lin
論文名稱:del Pezzo 曲面之幾何
論文名稱(外文):On Geometry of Del Pezzo Surfaces
指導教授:陳榮凱
口試委員:陳俊成江謝宏任余正道莊武諺
口試日期:2014-01-27
學位類別:博士
校院名稱:國立臺灣大學
系所名稱:數學研究所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2014
畢業學年度:102
語文別:英文
論文頁數:48
中文關鍵詞:del Pezzo 曲面奇點complement凱勒─愛因斯坦距離不消沒定理
外文關鍵詞:del Pezzo surfacessingularitiescomplementKahler-Einstein metricsnonvanishing
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本文介紹del Pezzo曲面之研究。早期的研究主要以光滑曲面為對向,但近年則多考慮帶有奇點的曲面。因此第二章即討論各種奇點,始自第三章起正式定義del Pezzo 曲面,介紹光滑曲面的分類。第四章介紹Shokurov發展的complement 理論,並在第五章的weighted complete intersection 中給出例子。第六章介紹凱勒─愛因斯坦距離和del Pezzo曲面的關係。第七章與第八章是作者的研究結果利用黎曼─羅赫定理計算尤拉示性數並得到一種特別的不消沒定理。

The thesis in on the geometry of del Pezzo surfaces. Early researches focused on smooth surfaces, while recently surfaces with singularities have been mostly considered. Consequently, in Chapter 2, different types of singularities are first discussed, and then del Pezzo surfaces can be defined formally in Chapter 3. Research on smooth surfaces are also given there. In Chapter 4, we introduce the complement theory developed by Shokurov, and we give some examples of weighted complete intersection in Chapter 5. Chapter 6 is about the relation between Kahler-Einstein metrics and del Pezzo surfaces. In Chapter 7 and Chapter 8, we introduce our research result. We use Riemann-Roch theorem to calculated Euler characteristics, and then give a special type of nonvanishing theorem.


目錄Contents
口試委員審定書i
誌謝ii
摘要iii
Abstract iv
1 Introduction 1
1.1 Notation and Conventions . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Singularities 2
2.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Log singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Toric varieties and singularities . . . . . . . . . . . . . . . . . . . . . 13
2.4 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Del Pezzo Surfaces 19
4 Complements on Log Surfaces 21
4.1 n-complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5 Weighted Complete Intersection 28
5.1 Weighted projective space . . . . . . . . . . . . . . . . . . . . . . . . 28
5.2 Weighted complete intersection . . . . . . . . . . . . . . . . . . . . . 30
6 Kahler-Einstein Metric 35
7 Euler Characteristics 37
7.1 Singular Riemann-Roch Theorem . . . . . . . . . . . . . . . . . . . . 37
7.2 Euler characteristics under L-blowups . . . . . . . . . . . . . . . . . . 40
8 Nonvanishing 43
Reference 46

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[16] Yu. G. Prokhorov.“ Lectures on complements on log surfaces”, arXiv:9912111.
[17] Yu. G. Prokhorov and A.B. Verevkin. “The Riemann-Roch theorem on surfaces
with log-terminal singularities” J. Math. Sci., 140, No.2 (2007).
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0 “. Invent, math. 89 (1987) 225-246.
[21] V. Tosatti, “Kahler-Einstein metrics on Fano surfaces”,Expositiones Mathematicae
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