Instructor: John Lesieutre
Meetings: MWF 2:00-2:50 (Stevenson Hall 104)
Email: jdl@uic.edu
Office hours: W 10-11, F 1-2, or just stop by (SEO 411)
Lecture | Date | Topic | Section | Vakil equivalent | |
Week 1: Sheaves. Homework (due W 1/20) | |||||
1 | 1/11 | Introduction to sheaves | §II.1 | Ch 2 | |
2 | 1/13 | Examples; morphisms of sheaves | §II.1 | Ch 2 | |
3 | 1/15 | Sheafification and other constructions | §II.1 | Ch 2 | |
Week 2: Schemes. Homework (due W 1/27) | |||||
4 | 1/20 | Spec of a ring | §II.2 | Ch 3 | |
5 | 1/22 | Affine schemes; other schemes | §II.2 | Ch 4.1-4.4 | |
Week 3: Schemes, continued. Homework (due F 2/5) | |||||
6 | 1/25 | Finish definitions, more examples | §II.2 | Ch 4.1-4.4; morphisms are Ch 6.1-6.3 | |
7 | 1/27 | The proj construction | §II.2 | Ch 4.5 | |
8 | 1/29 | Some conditions on schemes | §II.3 | Spread through Ch 3,5,6,7,8 | |
Week 4: More properties of schemes. Homework (due F 2/12) | |||||
9 | 2/2 | Open and closed embeddings, reduced induced structure | §II.3 | Ch 8.1, etc. | |
10 | 2/4 | Fiber products | §II.3 | Ch 9.1-9.3 | |
11 | 2/6 | Separated and proper morphisms/schemes | §II.4 | Ch 10 | |
Week 5: Sheaves of modules. Homework (due F 2/19) | |||||
12 | 2/8 | Separated and proper, part 2 | §II.4 | Ch 12.7 | |
13 | 2/10 | Finish properness; start O_X modules | §II.5 | Ch 13 | |
14 | 2/12 | More sheaves of modules, quasicoherence | §II.5 | Ch 13 | |
Week 6: Quasicoherent sheaves. Homework (due F 2/26) (#3) | |||||
15 | 2/15 | Basic properties of quasicoherent sheaves | §II.5 | 13.4 | |
16 | 2/17 | Closed subschemes; O(n) on Proj S | §II.5 | 14.1, though the approach is different | |
17 | 2/19 | Finish quasicoherent sheaves | §II.5 | 16.1-16.3 | |
Week 7: Divisors. Homework (due F 3/4) | |||||
18 | 2/22 | Weil divisors | §II.6 | 14.2 | |
19 | 2/24 | Cartier divisors and Pic(X) | §II.6 | 14.3 | |
20 | 2/26 | Finish up line bundles | §II.6 | ||
Week 8: Maps to projective space. Homework (due F 3/11) | |||||
21 | 2/29 | Invertible sheaves and maps to projective space | §II.7 | 14 | |
22 | 3/2 | Linear systems; relative proj | §II.7 | 14 | |
23 | 3/4 | Blowing up ideal sheaves, projectivization of locally free sheaves | §II.7 | 14 | |
Week 9: Differentials, etc. Homework (due F 3/18) | |||||
24 | 3/7 | Modules of Kahler differentials | §II.8 | 21.1 | |
25 | 3/9 | Sheaves of differentials, normal bundles | §II.8 | 21.2 | |
26 | 3/11 | Bertini's theorem, the canonical class | §II.8 | 12,25 | |
Week 10: Differentials, and some fun. Homework (due F 4/3) (and a Nakai criterion example) | |||||
27 | 3/14 | Canonical bundles and adjunction | §II.8 | 21.5 | |
28 | 3/16 | Kodaira dimension and the Enriques-Kodaira dimension | wiki | ||
29 | 3/18 | Some properties of ample line bundles | Lazarsfeld, Ch 1.2 | ||
Week 11: Cohomology Homework (due F 4/10) | |||||
30 | 3/28 | Derived functors | §III.1 | 23.2 | |
31 | 3/30 | Cohomology of sheaves | §III.2-3 | 23.4 | |
32 | 4/1 | Cech cohomology, part 1 | §III.4 | Ch 18 | |
Week 12: Cohomology, II Homework (due F 4/17) | |||||
33 | 4/4 | Cech cohomology, part 2 | §III.4 | 18 | |
34 | 4/6 | Cohomology of projective space | §III.5 | 18 | |
35 | 4/8 | Applications of Wednesday's calculation | §III.5 | 18 | |
Week 13: Ext and Serre duality Homework (due F 4/24) | |||||
36 | 4/11 | Euler characteristic | §III.5, Ex 1&2 | 18.2 | |
37 | 4/13 | Ext groups | §III.6 | 30.2 | |
38 | 4/15 | Serre duality, I | §III.7 | 30 | |
Week 14: Serre duality, flatness (no homework!) | |||||
39 | 4/18 | Serre duality, II + Kodaira vanishing | §III.7 | 30 | |
40 | 4/20 | Higher direct images | §III.8 | 18.8 | |
41 | 4/22 | Flatness | §III.9 | 24 | |
Week 13: Ext and Serre duality Homework (due F 4/24) | |||||
42 | 4/25 | Smooth morphisms | §III.10 | 25 | |
43 | 4/27 | Macaulay 2 | Problems, solutions | ||
44 | 4/29 | Something fun |