MODERN ALGEBRAIC GEOMETRY II, SPRING 2022, LECTURE AND HOMEWORK PLANS
[H]=Hartshorne
- Week 1, January 18 and 20. This week's classes will be held by Zoom.
Reading for the week. [H] Ch. III.1; Sections 1-4 of
Yekutieli's notes on derived categories
Lecture notes for the first week
Reading for Tuesday Jan. 25: [H] Ch. III.2.
Homework for Tuesday Feb 1 (this is an extension of the previously announced due date): [H] III.2.1-III.2.7
- Week 2, January 25 and 27. This week's classes, and all classes thereafter, will be held in person in 141 Altgeld Hall.
This week we will cover [H] Chapters III.2-3. As announced in class on Jan 25, the class on Jan 27 will begin with an invitation to work out homework problem III.3.1.
Reading for Tuesday Feb. 1: [H] Ch. III.4
Homework for Tuesday Feb 15: [H] Ch III, 3.1-5, 3.6(a), 3.7
- Week 3, February 1 and 3.
As mandated by campus, class on Thursday, February 3 will be held online due to winter weather. The Zoom link is the same one that was used during the first week of classes, and was sent to all registered students by email on February 1.
This week we will cover [H] Chapter 4 and begin [H] Chapter 5.
Class on Feb 1 will begin with a problem session covering [H] III.2.1-III.2.7.
Corrected lecture notes for Feb 3 class (taught online due to winter weather).
Reading for Tuesday Feb. 8: [H] Ch. III.5 and III.6
Additional homework for Tuesday Feb 15: [H] Chap III, 4.1, 4.3-4.5, 4.7
- Week 4, February 8 and 10.
This week we will finish [H] Chapter 4, cover [H] Chapter 5, and begin [H] Chapter 6.
Reading for Tuesday Feb. 15: [H] Ch. III.6 and III.7
Homework for Tuesday March 1: [H] Chap III, 5.2, 5.5-5.7, 5.10
- Week 5, February 15 and 17.
This week we will begin with a problem session on the problems from [H] Chapter 3 assigned during weeks 2 and 3. We will then
finish [H] Chapter 6 together with material on the composition of derived functors used to relate Ext and Ext_{O_X}.
We will begin [H] Chapter 7 if time allows.
Reference for derived categories: Weibel, An Introduction to Homological Algebra contains the result we need as Theorem
10.8.2, but this will be difficult to pick up and read if you don't already know about derived categories. If you want to learn this subject in depth, you will be rewarded by studying all of Chapter 10.
Reading for Tuesday Feb. 22: [H] Ch.III.7
Homework for Tuesday March 1: [H] Chap III, 6.1, 6.4, 6.7, 6.8
- Week 6, February 22 and 24.
We will
finish [H] Chapter 6 together with material on the composition of derived functors and spectral sequences used to relate Ext and Ext_{O_X}.
We will then begin [H] Chapter 7.
Reference for spectral sequences: Griffiths and Harris, Principles of Algebraic Geometry, Chapter 3.5. This reference does not include an application to Ext and Ext_{O_X}, but it includes the Leray spectral sequence (from a complex analytic perspective),
which is associated to the composition of the derived functors of the pushforward functor and the global section functor.
Reading for Tuesday March 1: [H] Chapter III.8
Homework for Tuesday March 22: [H] Chap III, 7.1-7.4
- Week 7, March 1 and 3.
This week we will begin with a problem session on the problems from [H] Chapter III.5 and III.6 assigned during weeks 4 and 5. We will then
finish [H] Chapter III.7 and begin [H] Chapter III.8
Reading for Tuesday March 8: [H] Chapter III.9
Homework for Tuesday March 22: [H] Chap III, 8.1-8.3
- Week 8, March 8 and 10.
This week we will
finish [H] Chapter III.7 and then cover [H] Chapter III.8
Reading for Tuesday March 22: [H] Chapter III.9
No additional Homework problems for Tuesday March 22.
- Week 9, March 22 and 24.
This week we will begin with a problem session on the problems from [H] Chapter III.7 and III.8 assigned during weeks 6 and 7. We will then
finish [H] Chapter III.8 and start [H] Chapter III.9
Reading for Tuesday March 29: [H] Chapters III.9, III.10
Homework for Tuesday, April 5: [H] Chap III, 9.1, 9.3-9.4
- Week 10, March 29 and 31.
This week we will finish [H] Chapter III.9 and start [H] Chapter III.10
Reading for Tuesday, April 5 : [H] Chapters III.10, III.11
Homework for Tuesday, April 5: [H] Chap III, 9.7-9.8 (the previously assigned 9.11 has been removed from the assignment)
- Week 11, April 5 and 7.
This week we will begin with a problem session on the problems in [H] Chapter III.9 assigned during weeks 9 and 10. Then we will finish [H] Chapter III.9 and cover [H] Chapter III.10.
Reading for Tuesday, April 12 : [H] Chapter III.11, III.12
Homework for Tuesday, April 19: [H] Chap III, 10.1-3, 10.5, 10.8
- Week 12, April 12 and 14.
This week we will finish [H] Chapter III.10, cover Chapter II.9 quickly, and cover Chapter III.11
Reading for Tuesday, April 19 : [H] III.12
Homework for Tuesday, April 19: [H] Chap III, 10.6, 11.1, 11.4
- Week 13, April 19 and 21.
This week we will begin with a problem session on the problems in [H] Chapters III.10 and III.11 assigned during weeks 11 and 12. Then we will quickly finish [H] Chapter III.11 and begin to cover [H] Chapter III.12 in detail.
Reading for Tuesday, April 26 : [H] V.1
Homework for Tuesday, May 3: [H] Chapter V, 1.3, 1.4
- Week 14, April 26 and 28.
This week we will start by finishing [H] Chapter III.12, and then go on to [H] Chapter V on surfaces beginning with V.1. Class on Tuesday, April 26 will be covered by Chris Dodd.
Some of the Chapter V material on surfaces relies on material on curves. While the relevant material on curves can be found in [H] Chapter IV and has been covered in other courses, I will not be assuming prior knowledge of the content of Chapter IV and will review/cover needed background on curves as it comes up rather than assigning reading from Chapter IV.
Reading for Tuesday, May 3 : [H] V.2
No additional homework for Tuesday, May 3.
- Week 15, May 3.
This week we will begin with a brief (no more than 30 minutes) problem session on the two problems from [H] Chapter V.1 assigned during week 13. Then in Chapter V.3 we will cover Props. 3.1-3.3 and 3.6 plus Cor. 3.7. We will then describe the famous 27 lines on a cubic surface (Prop 4.8 and Thm 4.9), omitting proofs. If time allows, we will then state the Castelnuovo contraction criterion (Theorem 5.7), with applications.