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2022　Advanced topics in Algebra H

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Academic unit or major
Graduate major in Mathematics
Instructor(s)
Kawachi Takeshi 
Class Format
Lecture    (Face-to-face)
Media-enhanced courses
Day/Period(Room No.)
Mon5-6(H102)  
Group
-
Course number
MTH.A504
Credits
1
Academic year
2022
Offered quarter
4Q
Syllabus updated
2022/4/20
Lecture notes updated
-
Language used
English
Access Index

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

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 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 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.

Out-of-Class Study Time (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

Assessment criteria and methods

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 (i.e., required knowledge, skills, courses, etc.)

No prerequisites are necessary, but enrollment in the related courses is desirable.

Other

None in particular.

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