課程資訊
課程名稱
代數幾何
Algebraic Geometry 
開課學期
106-2 
授課對象
理學院  數學系  
授課教師
林惠雯 
課號
MATH5146 
課程識別碼
221 U3600 
班次
 
學分
3.0 
全/半年
半年 
必/選修
選修 
上課時間
星期二3,4(10:20~12:10)星期四2(9:10~10:00) 
上課地點
天數102天數102 
備註
初選不開放。上課時間二10:20-11:50四08:30-10:00
總人數上限:60人 
Ceiba 課程網頁
http://ceiba.ntu.edu.tw/1062MATH5146_AlgGeo 
課程簡介影片
 
核心能力關聯
核心能力與課程規劃關聯圖
課程大綱
為確保您我的權利,請尊重智慧財產權及不得非法影印
課程概述

1. Scheme theory:
(a) Sheaves:the concept provides a systematic way of keeping track of local algebraic data on a topological space
(b) Schemes:they enlarge the category of varieties
(c) Important properties:integral、noetherian、finiteness、separatedness、properness
(d) Coherent sheaves:controllable sheaves on schemes

2. Intrinsic geometry:
(a) Linear system:an important tool for studying embedding problem
(b) Picard groups:important intrinsic invariants
(c) Sheaves of differential : algebraic version of differential forms
(d) Forma schemes : they carry information about all the infinitesimal neighborhoods

3. Cohomology theory:
We intend to define it by using derived functors and to compute it by introducing Cech cohomology. We also show some important vanishing theorems and combine them with Serre duality to give some applications in flat family、smooth family, including Zariski’s main theorem and base change theorem of cohomology.
 

課程目標
The main goal of this course is to introduce the Scheme theory which is the foundation for modern algebraic geometry and the Cohomology theory which is the major technique to define numerical invariants.  
課程要求
The background on "Commutative Algebra" or "Homological Algebra". 
預期每週課後學習時數
 
Office Hours
另約時間 
參考書目
D. Eisenbud and J. Harris : The Geometry of Schemes
Igor R. Shafarevich : Basic Algebraic Geometry 1 & 2
D. Mumford : The Red Book of Varieties and Schemes
Sergei I. Gelfand and Yuri I. Manin : Methods of Homological Algebra 
指定閱讀
R. Hartshorne : Algebraic Geometry 
評量方式
(僅供參考)
   
課程進度
週次
日期
單元主題
第1週
2/27,3/01  Sheaves / Affine schemes ; Chap II : 1.2, 1.7, 1.14, 1.16, 1.17, 1.19 / Chap II : 2.4, 2.7, 2.12 
第2週
3/06,3/08  Schemes / Noetherian property ; Chap II : 2.14, 2.16, 2.18 / Chap II : 3.2, 3.5, 3.7, 3.12, 3.14 
第3週
3/13,3/15  Separatedness (I) / Separatedness (II) ; Chap II : 3.13, 3.15, 3.17, 4.2 / Chap II : 4.7, 4.8 
第4週
3/20,3/22  Properness / Coherent sheaves (I) ; Chap II : 4.4, 4.6, 4.9 / Chap II : 5.1, 5.2, 5.6 
第5週
3/27,3/29  Coherent sheaves (II) / Serre theorem ; Chap II : 5.5, 5.7, 5.9 / Chap II : 5.10, 5.11, 5.13 
第6週
4/03,4/05  溫書假 / 掃墓節放假 
第7週
4/10,4/12  Divisors / Linear systems ; Chap II : 6.1, 6.2, 6.8 / Chap II : 7.1, 7.2, 7.5  
第8週
4/17,4/19  Proj F / Blow-ups ; Chap II : 5.15, 7.8, 7.10 / Chap II : 7.12, Proof of Theorem 7.17 
第9週
4/24,4/26  Differentials(I) / Differentials (II) ; Chap II : 8.1, Proof of Theorem 8.13 / Chap II : 8.2, 8.3 
第10週
5/01,5/03  Differentials (III) / Formal schemes (I) ; Chap II : 8.5, 8.8 / Chap II : 9.1, 9.3  
第11週
5/08,5/10  Formal scheme (II) / Cohomology of sheaves ; Chap II : 9.4 / Chap III : 2.3, 2.4, 2.6 
第12週
5/15,5/17  Cohomology of a noetherian affine scheme / Cech cohomology ; Chap III : 3.2, 3.3, 3.6 / Chap III : 4.4, 4.5, 4.7 
第13週
5/22,5/24  Cohomology of a projective space / Serre duality (I) ; Chap III : 5.5, 5.7, 5.10 / Chap III : 6.3, 6.10, 7.2  
第14週
5/29,5/31  Serre duality (II) / Flat morphisms (I) ; Chap III : 4.1, 6.7, 7.3 / Chap III : 5.2, 8.1, 9.1 
第15週
6/05,6/07  Flat morphisms (II) / Smooth morphisms ; Chap III : 9.2, 9.3, Proof of Lemma 9.12 / Chap III : 10.1, 10.2, 10.3 
第16週
6/12,6/14  Bertini theorem and Zariski's Main theorem / Lemma on formal functions 
第17週
6/19  Semicontinuity Theorem