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School of Environment and Society
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- Liberal arts and basic science courses
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Suzuki Masatoshi Kawachi Takeshi
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Mon5-6(H116)
- Group
- -
- Course number
- MTH.A508
- Credits
- 1
- Academic year
- 2017
- Offered quarter
- 4Q
- Syllabus updated
- 2017/3/17
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
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Course description and aims
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.
Student learning outcomes
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.
Keywords
Resolution of singularities, blowing-up, canonical singularity, terminal singularity, rational singularity, minimal model.
Competencies that will be developed
| ✔ 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/Required learning
| 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 |
Textbook(s)
None in particular.
Reference books, course materials, etc.
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
Assessment criteria and methods
Assessments on reports (100%).
Related courses
- MTH.A201 ： Introduction to Algebra I
- MTH.A202 ： Introduction to Algebra II
- MTH.A301 ： Algebra I
- MTH.A302 ： Algebra II
- MTH.A507 ： Advanced topics in Algebra G1
Prerequisites (i.e., required knowledge, skills, courses, etc.)
No prerequisites are necessary, but enrollment in the related courses is desirable.
