Algebraic geometry III - selected topics
Summary
This course is aimed to give students an introduction to the theory of algebraic curves and surfaces. In particular, it aims to develop the students' geometric intuition and combined with the basic algebraic geometry courses to build a strong foundation for further study.
Content
- Separated and proper morphism, varieties using the language of schemes
- Recap: Divisors, sheaf cohomology and morphisms to projective spaces
- Riemann-Roch and Serre duality for curves
- Classification of curves
- Embedding of curves in projective spaces
- Algebraic surfaces
- Intersection theory on smooth surfaces
- Blow-ups
- Fibrations of surfaces
Keywords
Algebraic geometry, curves, surfaces, singularities, birational geometry
Learning Prerequisites
Required courses
- Linear algebra
- Group Theory
- Rings and Modules
- Modern Algebraic geometry
Recommended courses
- Topology I & II
- Algebraic topology
- Differential geometry
- Algebraic number theory
- Schemes
- Complex manifolds
- Complex Analysis
Learning Outcomes
- Analyze basic problems in algebraic geometry of curves and surfaces and solve them.
- Recall the statements of basic theorems like Riemann-Roch, the Hodge index theorem, Castelnuovo's criteria, etc., and understand their proofs
- Compute geometric and birational invariants of curves and surfaces in basic examples.
- Formulate a sketch of the birational classification of surfaces and how to approach its proof.
- Reason intuitively about curves and surfaces over the complex numbers.
Teaching methods
2h lectures+2h exercise sessions weekly.
Assessment methods
Oral Exam
Supervision
Office hours | Yes |
Assistants | Yes |
Forum | No |
Resources
Bibliography
We will follow mainly
- Hartshorne, Algebraic Geometry
- Liu, Algebraic Geometry and Arithmetic Curves
- Beauville, Complex Algebraic Surfaces
Other resources students may want to look at are
- R. Miranda, Algebraic Curves and Riemann Surfaces
- M. Reid, Chapters on Algebraic Surfaces
Ressources en bibliothèque
- Chapters on Algebraic Surfaces / Reid
- Algebraic Curves and Riemann Surfaces / Miranda
- Algebraic Geometry and Arithmetic Curves / Liu
- Algebraic Geometry / Hartshorne
- Algebraic Curves and Riemann Surfaces / Miranda
- Complex Algebraic Surfaces / Beauville
Références suggérées par la bibliothèque
Moodle Link
Dans les plans d'études
- Semestre: Printemps
- Forme de l'examen: Oral (session d'été)
- Matière examinée: Algebraic geometry III - selected topics
- Cours: 2 Heure(s) hebdo x 14 semaines
- Exercices: 2 Heure(s) hebdo x 14 semaines
- Semestre: Printemps
- Forme de l'examen: Oral (session d'été)
- Matière examinée: Algebraic geometry III - selected topics
- Cours: 2 Heure(s) hebdo x 14 semaines
- Exercices: 2 Heure(s) hebdo x 14 semaines
- Semestre: Printemps
- Forme de l'examen: Oral (session d'été)
- Matière examinée: Algebraic geometry III - selected topics
- Cours: 2 Heure(s) hebdo x 14 semaines
- Exercices: 2 Heure(s) hebdo x 14 semaines
Semaine de référence
Lu | Ma | Me | Je | Ve | |
8-9 | |||||
9-10 | |||||
10-11 | |||||
11-12 | |||||
12-13 | |||||
13-14 | |||||
14-15 | |||||
15-16 | |||||
16-17 | |||||
17-18 | |||||
18-19 | |||||
19-20 | |||||
20-21 | |||||
21-22 |
Légendes:
Cours
Exercice, TP
Projet, autre