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- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Takagi Shunsuke Shimomoto Kazuma
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- Day/Period(Room No.)
- Intensive ()
- Group
- -
- Course number
- MTH.E643
- Credits
- 2
- Academic year
- 2024
- Offered quarter
- 2Q
- Syllabus updated
- 2024/3/14
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
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Course description and aims
"F-singularities" refer to singularities defined by the Frobenius map. There are four basic classes of F-singularities: F-regular, F-pure, F-rational, and F-injective. These are expected to correspond to major classes of singularities in complex birational geometry. In this course, I will outline the recent development of this correspondence.
Student learning outcomes
The aim is to learn and become accustomed to various concepts that appear in the singularity theory of algebraic varieties, with a focus on positive characteristic methods.
Keywords
F-singularities, BCM test ideals, multiplier ideals, reduction modulo p
Competencies that will be developed
| ✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Class flow
This is a standard lecture course. There will be some assignments.
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | The following topics will be covered in this order : -- Basic notions in the theory of F-singularities (F-regular, F-pure, F-rational, and F-injective singulariteis) -- Singularities in birational geometry (log terminal, log canonical, rational, and Du Bois singularities) -- Absolute integral closures and BCM test ideals -- Reduction modulo p technique due to Deligne-Illusie-Raynaud -- Weak ordinarity conjecture | Details will be provided during each class session |
Textbook(s)
None required
Reference books, course materials, etc.
S. Takagi and K.-i. Watanabe, F-singularities: applications of characteristic p methods to singularity theory, Sugaku Expositions 31 (2018), no.1, 1–42.
Assessment criteria and methods
Assignments (100%).
Related courses
- MTH.A301 ： Algebra I
- LAS.M106 ： Linear Algebra II
- MTH.A331 ： Algebra III
Prerequisites (i.e., required knowledge, skills, courses, etc.)
None required
