Algebraic Geometry III
SF3606 Algebraic Geometry, Fall 2012, 7.5 hp.
Course structure
The course is given as a series of lectures, consisting of two hours every week. The course starts on Thursday Sep 6, 13-15 in 3733. Starting with Oct 23, we will run on Tuesdays 10-12 in 3733 (with the exception of Nov 8). The course will run as a reading course where the participants are expected to lecture.
Contents
Čech cohomology of quasi-coherent sheaves on schemes, applications to curves (Riemann-Roch, Riemann-Hurwitz), differentials, derived functors and flatness.
Course material
We will mainly use Ravi Vakil's notes for Math 216 (Foundations Of Algebraic Geometry). We will use the May 16, 2012 version . We will focus on part V (sections 12-25). And we use Hartshorne "Algebraic Geometry" chapter 3.
Prerequisites
Knowledge of basic algebraic geometry (schemes, sheaves, etc.) on the level of Algebraic Geometry II (FSF3605) given Spring 2012.
Course plan (Preliminary)
| # | Date, Time, Place | Topics | Key results* | Speaker | Exercises |
|---|---|---|---|---|---|
| 1 | Sept 6, 13.15-15.00, 3733 | 14.5, 17.7: Q-coh sheaves, representable functors, Grassmannians | Roy (x2) | Plucker embedding (pdf 116 kB) | |
| 2 | Sept 13 | (14.1), 16.3, 17.4, 17.6: Invertible sheaves, global generation, base-point freeness, ample and avery ample sheaves | Thms 16.3.1, 16.3.8, 17.4.1, 17.6.2 | David (x2) | Veronese, Segre, very ample, ample (pdf 71 kB) |
| 3 | Sept 20 | 20.1: Properties of cohomology | David | ||
| 20.2.1-4: Cech cohomology | Thms 20.2.2, 20.2.4 and 20.2.6 | Theo | Exercises 20.2: A, E, F | ||
| 4 | Sept 27 | 20.3: Cohomology of line bundles ... | Jared | Exercise 20.3.E | |
| 20.4: Riemann Roch | Anders | Exercise 20.4.A | |||
| 5 | Oct 4 | 20.4.4: Serre Duality | Olof | Exercise 20.4.M | |
| 20.5: Hilbert polynomial | Matheus | Prime avoidance and Hilbert polynomials (pdf 90 kB) | |||
| 6 | Oct 11 | 20.6: Serre's vanishing criteria | Thm 20.6.1 and 20.6.6 | Gustav | Exercise 20.6.A/B |
| 20.7: Higher direct images | Thm 20.7.1 and 20.7.B | David | Exercise 20.7.F | ||
| 7 | Oct 18 | 20.8: Chow's Lemma and Coherence Thm | Thms 20.8.1 and 20.8.2 | Sebastian | Exercise 20.8.1.A, B. |
| 21.1: Criterion for being a closed immersion | Thm 21.1.1 | Alessandro | |||
| 8 | Oct 23 10.15-12.00 |
21.2: Toolbox for curves | 21.2.10, 21.2.11 and 21.2.C | Katharina | |
| 21.3: Curves of genus 0 | 21.3.1 and 21.3.2 | Katharina | Twisted cubic (pdf 94 kB) | ||
| Oct 30 | Break | ||||
| 9 | Nov 8 13.15-15.00 |
Elliptic curves | Carel (x2) | ||
| 10 | Nov 13 | 23.1-23.2.16: Differentials | Gustav (x2) | AGIIIExc13nov.pdf (pdf 90 kB) | |
| 11 | Nov 20 | 23.2.17-23.2.Q: First properties | Anders | Differentials (pdf 109 kB) | |
| 23.3-23.4.D: Cotangent bundle | Sebastian | ||||
| 12 | Nov 27 | 21.5: Hyperelliptic curves | Olof | ||
| 23.4.6-23.4.9, 23.5-? | Alessandro | ||||
| 13 | Dec 4 | 23.5.?-23.5.Q | Alessandro | ||
| H III.1-2: Derived functors | Jared | ||||
| 14 | Dec 11 | H III.3: Cohomology of affine | Olof | ||
| H III.4: Derived cohomology equals Čech cohomology. | Anders | ||||
* Key results that should be presented during the lecture.
Exercises to lectures 1-3 to be handed in Oct 11.
Exercises to lectures 4-5 to be handed in Oct 23.
Exercises to lectures 6-8 to be handed in Nov 13.
Exercises to lectures 10-11 to be handed in Dec 4.