Algebraic Geometry
SF3605 Algebraic Geometry II, spring 2012, 7.5 hp.
Course structure
The course is given as a series of lectures, consisting of two hours every week. The first lecture is given Tuesday, January 31st. Every second week there will be exercise sessions, where the students are expected to present and discuss exercises.
- Lecturer: Roy Skjelnes.
- Tuesdays 10-12, room 3733
- Exercise sessions; Selected Wednesdays, 09-11, room 3721.
Contents
Sheaves; coherent sheaf of modules, invertible sheaves, bundles, Kähler differentials. Schemes; open and closed subschemes, affine schemes, projectives schemes, varieties, Morphisms; separated, proper, projective and affine morphisms. Divisors, Grassman, linear systems, blow-up.
If time permits: Functor of points, representability, etale morphisms, descent.
Course material
We will mainly use Algebraic Geometry by Robin Hartshorne, chapters 2.1-2.8. But, we will also rely on The Red book of Varieties and Schemes by David Mumford.
Prerequisites
Knowledge of topology and commutative algebra is needed, for instance SF2735 and SF2737. We will assume that the participant is familiar with the concepts: Topological spaces, cover, compactness, commutative rings, prime and maximal ideals, localization of rings, modules.
Examination
Home assignments and participation in the exercise sessions.
Home assignment
1. The first home assignment is to do Exercise 3.11, Chapter 2.3. Deadline is March 20.
2. The second assignment is Exercsi 5.17, Chapter 2.5. Deadline is April 24.
Course Plan
The following course plan, including suggestions for exercises, will be regularly updated.
AM refers to Atiyah-MacDonald, Introduction to Commutative Algebra.
| Date | Theme | Exercises |
|---|---|---|
| 31.01 | Sheaves (pp. 60-64) (pdf 203 kB) | 2.1: 1.6 1.7, 1.8, 1.14, 1.15, 1.18, 1.19. |
| 07.02 | Affine schemes (pp. 65-71) (pdf 218 kB) | AM, ch. 1: 15, 16, 17, 18, 19, 20, 21. |
| 14.02 | Schemes (pp 72-75) (pdf 200 kB) | 2.2: 2.3, 2.7, 2.8, 2.13, 2.14, 2.16, 2.17, 2.18. |
| 21.02 | Subschemes (pp 75-82, 85) (pdf 216 kB) | 2.3: 3.6, 3.18 |
| 22.02 | Exercise session. | Olof, Theo |
| 28.02 | Fiber product (pp.82-95) (pdf 190 kB) | 2.3: 3.3, 3.4, 3.5, 3.9, 3.10, 3.11. |
| 06.03 | Closed maps (pdf 199 kB) | 2.4: 4.1, 4.2, 4.3, 4.4, |
| 07.03 | Exercise session. | Mihai, Fabian, Gustav, Magnus |
| 13.03 | q-coherent sheaves (pp 108-114) (pdf 224 kB) | 2.5: 5.2, 5.3, 5.6, 5.7, 5.8 |
| 20.03 | q-coherent sheaves II (pp. 113-116) (pdf 217 kB) | 2.5: 5.1, 5.4, 5.5, 5.15, 5.16, 5.17, 5.18. |
| 21.03 | Exercise session | Anders, Erik, Jared |
| 27.03 | Kähler differentials (pdf 236 kB) | 2.8. 8.6 |
| 03.04 | O(1) (pdf 225 kB) | |
| 04.04 | Exercise session | Gustav, Mihai, Olof |
| 17.04 | Projective results (pdf 209 kB) | 1.1: 1.2, 1.3 1.2: 2.13, 1.3: 3.2 |
| 18.04 | Exercise session | Sebastian (3.18) (pdf 0 kB) , Linus, Mateus, Magnus |
| 24.04 | Weil divisors (pdf 198 kB) | |
| 08.05 | Cartier divisors (pdf 218 kB) | |
| 15.05 | Projective maps (pdf 183 kB) |