Math 553

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)

Syllabus and course info

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



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