2016 Advanced topics in Geometry G

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Academic unit or major
Graduate major in Mathematics
Instructor(s)
Honda Nobuhiro  Kawai Shingo 
Course component(s)
Lecture     
Day/Period(Room No.)
Fri5-6(H119A)  
Group
-
Course number
MTH.B503
Credits
1
Academic year
2016
Offered quarter
3Q
Syllabus updated
2016/9/27
Lecture notes updated
-
Language used
Japanese
Access Index

Course description and aims

In this course we will discuss some selected topics on linear differential equations called Fuchsian projective connections on complex algebraic curves. The notion of a Fuchsian projective connection has quite a long history; it appears, for instance, in the study of uniformization of complex algebraic curves by Poincare. While there are many relevant topics, we will focus on degenerations of (four, in particular) pointed projective lines into pointed stable rational curves, and their interplay with Fuchsian projective connections. Degenerations of 4-pointed projective lines are one of the most elementary degeneration processes in algebraic geometry; however, we will find some non-trivial things happen when simultaneously dealing with families of Fuchsian projective connections with regular singularities at the 4 marked points on the degenerating lines. Along the way we will try to get across some of the main ideas in the constructions of various moduli spaces while doing lots of calculations.

Student learning outcomes

On successful completion of this course, students will be able to:
(1) Understand properly and (in the case of 4-pointed curves) describe explicitly the moduli space of pointed stable rational curves and the universal family.
(2) Understand properly and describe correctly the (relative) dualizing sheaves of (families of) singular curves and the direct images of their tensor powers.
(3) Understand properly and describe correctly the (relative) projective structures and (relative) Fuchsian projective connections on (families of) singular curves.
(4) Understand properly and describe explicitly the moduli space of Fuchsian projective connections on 4-pointed stable rational curves.

Keywords

(Relative) projective structure, (relative) Fuchsian projective connection, (relative) Schwarzian derivative, pointed projective line, pointed stable rational curve, moduli space, universal family, (relative) dualizing sheaf, direct image sheaf

Competencies that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

To help students' understanding of the course material, problem sets and references will be provided throughout the course.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Introduction---Motivating example Details will be provided in each class session.
Class 2 (Relative) projective structures and (relative) Fuchsian projective connections on (families of) nonsingular curves Details will be provided in each class session.
Class 3 Degenerations of pointed projective lines, moduli spaces of pointed stable rational curves and the universal families Details will be provided in each class session.
Class 4 (Relative) dualizing sheaves of (families of) singular curves and the direct images of their tensor powers Details will be provided in each class session.
Class 5 (Relative) projective structures and (relative) Fuchsian projective connections on (families of) singular curves Details will be provided in each class session.
Class 6 Relative Schwarzian derivative and perfect bases for certain second-order linear partial differential equations Details will be provided in each class session.
Class 7 Moduli space of Fuchsian projective connections on 4-pointed stable rational curves Details will be provided in each class session.
Class 8 Directions for further research Details will be provided in each class session.

Textbook(s)

None required.

Reference books, course materials, etc.

While methods and setups are quite different, the following two references deal with relevant issues:
Mochizuki, Shinichi, A theory of ordinary p-adic curves, Publ. Res. Inst. Math. Sci. 32 (1996), 957--1152.
Mochizuki, Shinihci, Foundations of p-adic Teichmuller theory, AMS/IP Studies in Advanced Mathematics 11, American Mathematical Society; International Press, 1999.
As for the notion of perfect bases for linear differential equations, see the following reference:
Gann, Sebastian and Hauser, Herwig, Perfect bases for differential equations, J. Symbolic Comput. 40 (2005), 979--997.

Assessment criteria and methods

The course grade will be based on problem sets assigned throughout the course.

Related courses

  • None in particular.

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

No prerequisites.

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