2022 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Algebra H
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Takeshi Kawachi
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-6 Mon (H102)
- Class
- -
- Course Code
- MTH.A504
- Number of credits
- 100
- Course offered
- 2022
- Offered quarter
- 4Q
- Syllabus updated
- Jul 10, 2025
- Language
- English
Syllabus
Course overview and goals
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 description and aims
Students are expected to:
- Find a resolution of singularities and determine its strict transformation and the exceptional divisor.
- Classify the type of singularities.
- Understand the differences of each singularities.
Keywords
Resolution of singularities, blowing-up, canonical singularity, terminal singularity, rational singularity, minimal model.
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Ordinary lectures. Assignments will be given during class sessions.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Blowing-ups | Details will be provided during each class session. |
| Class 2 | Resolution of singularities | Details will be provided during each class session. |
| Class 3 | Canonical and terminal singularities | Details will be provided during each class session. |
| Class 4 | Resolution of plane singularities | Details will be provided during each class session. |
| Class 5 | Rational singularities | Details will be provided during each class session. |
| Class 6 | Non rational singularities | Details will be provided during each class session. |
| Class 7 | Minimal model | Details will be provided during each class session. |
| Class 8 | Minimal model | Details will be provided during each class session. |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
None in particular.
Reference books, course materials, etc.
Shihoko Ishii, "Introduction to Singularities", Springer-Verlag, ISBN 978-4-431-56261-0
Yuriro Kawamata, "Algebraic Variety Theory", Kyoritsu Shuppan Co. Ltd. ISBN 4-320-01571-1
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
Evaluation methods and criteria
Learning achievement is evaluated by reports(100%).
Related courses
- MTH.A301 ： Algebra I
- MTH.A302 ： Algebra II
- MTH.A507 ： Advanced topics in Algebra G1
Prerequisites
No prerequisites are necessary, but enrollment in the related courses is desirable.
Other
None in particular.