2022 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Algebra G
- 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.A503
- Number of credits
- 100
- Course offered
- 2022
- Offered quarter
- 3Q
- Syllabus updated
- Jul 10, 2025
- Language
- English
Syllabus
Course overview and goals
In this course the major subjects of algebraic geometry such as sheaves, divisors and related topics are treated. Divisors and its associated sheaves, and the canonical sheaf and the canonical divisor defind from algebraic varieties are displayed. Then properties the algebraic variety has might be determined by divisors and its intersection number.
Based on ring theory and modules, algebraic varieties and sheaves are defined. The geometric properties as manifold are expressed in algebraic terms, also the algebraic properties are characterized geometrically. Mathematical relationship between algebraic and geometric subjects will be acquired through this course.
Course description and aims
Students are expected to:
- Understanding the sheaves and the divisors, and could find the direct and inverse image of sheaves and divisors,
- Calculate the intersection number of divisors,
- Understanding the canonical sheaf and the canonical divisor, and could determine the ramification divisor.
Keywords
Algebraic variety, sheaf, divisor, singularity.
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 | Algebraic sets and rational maps | Details will be provided during each class session. |
| Class 2 | Sheaves | Details will be provided during each class session. |
| Class 3 | Algebraic varieties and their maps | Details will be provided during each class session. |
| Class 4 | Divisors and locally free sheaves | Details will be provided during each class session. |
| Class 5 | The canonical sheaf | Details will be provided during each class session. |
| Class 6 | Intersection numbers | Details will be provided during each class session. |
| Class 7 | Singularities | Details will be provided during each class session. |
| Class 8 | Algebraization theorem | 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.
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.