NCTS/TPE Course on

Algebraic and Arithmetic Geometry

代數幾何與算術幾何

Course Information

Instructor : Jungkai Chen陳榮凱, I-Chau Huang黃一樵, Chia-Fu Yu余家富

Place: New Math Bldg 102

Hours: Tue.9:30-11:00, 11:10-12:10

 

Syllabus

The purpose of this course is to provide foundation knowledge for students who are interest in algebraic and arithmetic geometry.

It will be a mixture of three components:

1. lectures/ seminar: base on Hartshorne's Algebraic Geometry.

2. problem session.

3. topics/ mini course: some topics such as

        1. Chevalley's Theorem on constructibility

        2. Normal schemes and normalization

        3. Excellent rings and Nagata rings

        4. Zariski's Main Theorem

Each meeting will basically be 1+1/2 hours of lecture plus 1 hour discussion of supplementary material and/or problems. Besides, at some point, there are going to have some topical talks related to the given lectures.

 

Topics for coming weeks:

        Oct. 5      Lecture: Dimension theory and Hilbert function

Discussion: Segre embedding, d-uple embedding and some examples of varieties.

Lecturer: Jungkai Chen

        Oct. 12    Lecture: Morphism and birational map of varieties

                        Discussion: Topology of rings

                        Lecturer: Jungkai Chen

        Oct. 19    Lecture: Singularities, regular local rings and factorial varieties

                        Discussion: Multiplicity.

                        Lecturer: Jungkai Chen

Oct. 26    Lecture: One-dimensional regular local rings, intersections in projective varieties

                        Discussion: Some examples

                        Lecturer: Jungkai Chen

        Nov. 2      Lecture: Affine scheme

                        Lecturer: I-Chau Huang

        Nov. 9      Lecture: Further properties of schemes.

                        Lecturer: I-Chau Huang

        Nov. 16    Lecture: Separated and proper morphism

                        Lecturer: I-Chau Huang

        Nov 23    Lecture: Divisors

                        Lecturer: I-Chau Huang

        Nov. 30   Lecture: Differentials

                        Lecturer: I-Chau Huang

        Dec. 7      Lecture: Chevalley's Theorem on constructability

                        Lecturer: Chia-Fu Yu

        Dec. 14    Lecture: Normal schemes and normalization

                        Lecturer: Chia-Fu Yu

        Dec. 21   Lecture: Excellent rings and Nagata rings

                        Lecturer: Chia-Fu Yu

        Dec. 28    Lecture: Zariski's Main Theorem

                        Lecturer: Chia-Fu Yu

        Jan. 4       Lecture: What are algebraic geometry and arithmetic geometry?

                        Lecturer: Chia-Fu Yu

 

Prerequisite

Algebra. But of course, it will involve a lot of commutative algebra.

 

Reference:

R. Hartshorne, Algebraic Geometry

D. Mumford, The Red Book of Varieties and Schemes