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2019　Advanced topics in Algebra G1

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Academic unit or major
Graduate major in Mathematics
Instructor(s)
Wakabayashi Yasuhiro 
Class Format
Lecture     
Media-enhanced courses
Day/Period(Room No.)
Mon5-6(H102)  
Group
-
Course number
MTH.A507
Credits
1
Academic year
2019
Offered quarter
3Q
Syllabus updated
2019/3/18
Lecture notes updated
-
Language used
English
Access Index

Course description and aims

This course focuses on the theory of stacks (which are natural extensions of the notion of scheme), and introduces the moduli space of (algebraic) curves.
The concept of “moduli” as parameters in some sense measuring or describing the variation of geometric objects was used in algebraic geometry, but it was not until the 1960s that D. Mumford and others gave precise definitions of moduli spaces and methods for constructing them. Since then there has been an enormous amount of work on and using moduli spaces from many different points of view. Also, the theory of stacks has its origins in the study of moduli spaces in algebraic geometry.
This course aims to establish a base on which students can pursue research in areas that use stacks and moduli spaces of curves, and to proceed on to more advanced topics through other studies. (But, there is a possibility that course schedule may be modified.)

Student learning outcomes

By the end of this course, students will be able to:
(1) Understand the concept of “moduli” through examples.
(2) Obtain basic knowledge of stacks and the moduli space of curves.

Keywords

Grothendieck topologies, descent, (Deligne-Mumford) stacks, moduli of algebraic curves, stable curves, the Deligne-Mumford compactification.

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 Introduction Details will be provided during each class session.
Class 2 Grothendieck topologies and fibered categories
Class 3 Descent and stacks
Class 4 Basics of stacks
Class 5 Deligne-Mumford stacks
Class 6 Moduli of curves
Class 7 Stable curves and the Deligne-Mumford compactification I
Class 8 Stable curves and the Deligne-Mumford compactification II

Textbook(s)

None in particular.

Reference books, course materials, etc.

Handouts will be distributed at the beginning of class when necessary.

Assessment criteria and methods

Assessments on reports.

Related courses

  • MTH.A201 ： Introduction to Algebra I
  • MTH.A202 ： Introduction to Algebra II
  • MTH.A301 ： Algebra I
  • MTH.A302 ： Algebra II

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

Basic knowledge of scheme theory (e.g., R. Hartshorne, "Algebraic Geometry", GTM 52, Springer-Verlag, ISBN 0-387-90244-9).

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