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.