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課程名稱 |
極小模型與奇點理論導論
Introduction to the Minimal Model Program and Singularities |
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開課學期 |
112-1 |
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授課對象 |
理學院 數學研究所 |
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授課教師 |
王賜聖 |
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課號 |
MATH5292 |
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課程識別碼 |
221 U9340 |
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班次 |
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學分 |
4.0 |
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全/半年 |
半年 |
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必/選修 |
選修 |
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上課時間 |
星期一3,4(10:20~12:10)星期四3,4(10:20~12:10) |
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上課地點 |
天數201天數201 |
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備註 |
總人數上限：20人 |
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課程簡介影片 |
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核心能力關聯 |
本課程尚未建立核心能力關連 |
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課程大綱
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課程概述 |
This course will introduce the Minimal Model Program (MMP), including singularities in MMP and the cone theorem. We will also study 3-dimensional terminal and canonical singularities in more details. Topics will include:
• Mori’s existence theorem of rational cuves
• Singularities in MMP
• Cone theorems
• Elliptic surface singularities and 3-fold canonical singularities
• Classification of 3-fold terminal singularities
• Existence of 3-fold flips (after Shokurov) |
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課程目標 |
The goal of this course is to provide general knowledge and skills in birational geometry. |
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課程要求 |
Knowledge of algebraic geometry is required:
• R. Hartshorne, Algebraic Geometry.
• A. Beauville, Complex algebraic surfaces.
Preliminary knowledge of the theory of deformation will be very helpful.
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預期每週課後學習時數 |
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Office Hours |
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指定閱讀 |
J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, 1998 |
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參考書目 |
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評量方式
(僅供參考) |
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No. |
項目 |
百分比 |
說明 |
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1. |
作業 |
60% |
Two homework assignments a week |
2. |
期末報告 |
40% |
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週次 |
日期 |
單元主題 |
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第1週 |
9/4, 9/7 |
Motivation of the Mori program, and cone of divisors |
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第2週 |
9/11, 9/14 |
Existence of rational curves |
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第3週 |
9/18, 9/21 |
Kawamata–Viehweg Vanishing theorem |
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第4週 |
9/25, 9/28 |
Singularities in the MMP |
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第5週 |
10/2, 10/5 |
Cone theorem: the Base point free theorem, Rationality theorem |
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第6週 |
10/12 |
Cone theorem: Kawamata's result on lengths of extremal rays |
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第7週 |
10/16, 10/19 |
The (Artin) fundamental cycle and Du Val surface singularities |
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第8週 |
10/23, 10/26 |
Laufer–Reid's Theorem for elliptic Gorenstein surface singularities |
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第9週 |
10/30, 11/2 |
Rational singularities and ramified covers |
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第10週 |
11/6, 11/9 |
Canonical and terminal 3-fold singularities of index one |
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第11週 |
11/13, 11/16 |
Quotient singularities and terminal lemma |
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第12週 |
11/20, 11/23 |
Mori's classification of terminal 3-fold singularities |
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第13週 |
11/27, 11/30 |
An introduction to flips and flops, and terminal 3-fold flops |
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第14週 |
12/4, 12/7 |
3-fold flips after Shokurov: pl flipping contractions and b-divisors |
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第15週 |
12/11, 12/14 |
3-fold flips after Shokurov: finite generation conjectue on surfaces |
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