In this course the classification of singularities and their properties are treated. A resolution of singularities and birational transformation to a minimal model are described.
Unlike smooth manifolds algebraic varieties usually contain singular locus. The exceptional divisor of a resolution contains information of singularities, it helps to understanding the character of singularities.
Course goals are:
Find a resolution of singularities and determine its strict transformation and the exceptional divisor.
Classify the type of singularities.
Understanding the differences of each singularities.
Resolution of singularities, blowing-up, canonical singularity, terminal singularity, rational singularity, minimal model.
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Ordinary lectures. Assignments will be given during class sessions.
Course schedule | Required learning | |
---|---|---|
Class 1 | Blowing-ups | Details will be provided during each class session. |
Class 2 | Resolution of singularities | |
Class 3 | Canonical and terminal singularities | |
Class 4 | Resolution of plane singularities | |
Class 5 | Rational singularities | |
Class 6 | Non rational singularities | |
Class 7 | Minimal model | |
Class 8 | Minimal models |
None in particular.
Shihoko Ishii, "Introduction to Singularities", Springer-Verlag, ISBN 978-4-431-56261-0
Igor R. Shafarevich, "Basic Algebraic Geometry 1", Springer, ISBN 978-3-642-42726-8
Igor R. Shafarevich, "Basic Algebraic Geometry 2", Springer, ISBN 978-3-662-51401-6
Robin Hartshorne, "Algebraic Geometry", GTM 52, Springer-Verlag, ISBN 0-387-90244-9
Assessments on reports (100%).
No prerequisites are necessary, but enrollment in the related courses is desirable.