Algebraic geometry III - selected topics
Summary
This course is an introduction to the theory of algebraic curves and surfaces. An important aim of the course is to develop geometric intuition while using the language of schemes developed in the basic algebraic geometry course, thus building a solid foundation for further study.
Content
- Recap: Divisors, sheaf cohomology and morphisms to projective spaces
- Riemann-Roch and Serre duality for curves
- Riemann-Hurwitz
- Classification of curves
- Embedding of curves in projective spaces
- Algebraic surfaces
- Intersection theory on smooth surfaces
- Blow-ups
- Fibrations of surfaces
Keywords
Algebraic geometry, algebraic curves, algebraic surfaces
Learning Prerequisites
Required courses
- Linear algebra
- Group Theory
- Rings and Modules
- Modern Algebraic geometry
Recommended courses
- Algebraic topology
- Differential geometry
- Complex Analysis
Learning Outcomes
By the end of the course, the student must be able to:
- Analyze basic problems in algebraic geometry of curves and surfaces and solve them.
- Use the statements of basic theorems like Riemann-Roch, Serre duality, etc, and understand their proofs.
- Reason intuitively about the geometry and topology of curves over the complex and finite fields.
- Compute geometric and birational invariants of curves and surfaces in basic examples.
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
- Vakil, The Rising Sea: Foundations of Algebraic Geometry
- Görtz-Wedhorn, Algebraic Geometry I & II
Ressources en bibliothèque
Références suggérées par la bibliothèque
Moodle Link
In the programs
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Algebraic geometry III - selected topics
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Algebraic geometry III - selected topics
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Algebraic geometry III - selected topics
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional