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.)
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.
Grothendieck topologies, descent, (Deligne-Mumford) stacks, moduli of algebraic curves, stable curves, the Deligne-Mumford compactification.
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Ordinary lectures. Assignments will be given during class sessions.
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 |
None in particular.
Handouts will be distributed at the beginning of class when necessary.
Assessments on reports.
Basic knowledge of scheme theory (e.g., R. Hartshorne, "Algebraic Geometry", GTM 52, Springer-Verlag, ISBN 0-387-90244-9).