To understand geometry of algebraic curves from the view point of compact Riemann surfaces.

Applied Complex Variables (Math 210)
Modern algebra 1, 2. (Math 301, 302)
General topology (Math 321)

Midterm 30%
Final exam 40%
Homework and Miscellanies: 30%

Frances Kirwan, Complex algebraic curves, Cambridge University Press/1992

1st Week
1.Riemann sphere
1.1 Projective line as a set of ratios
1.2 Topology of the projective line
1.3 Projective line by gluing two complex planes

2nd Week
1.4 Projective line as a comfactification of the complex plane
1.5 Coordinate systems in the projective line
1.6 Regular and rational functions on the projective line and its open sets
1.7 Regular self-maps of the projective line

3rd Week
2. Riemann surface2.1 Definition of Riemann surfaces
2.2 Regular maps of Riemann surfaces

4th Week
2.3 Regular maps of compact Riemann surfaces
3. Differentials on Riemann surfaces
3.1 Tangent spaces and cotangent spaces

5th Week
3.2 Differential 1-form
3.3 Differential 2-form
3.4 Exterior derivatives of differentials
3.5 Pull-backs of differrntials

6th Week
3.6 Branch and rational differential
3.7 Integration
4. Various Riemann spheres
4.1 Complex manifold
4.2 One-dimensional tori

7th Week
4.3 Affine plane curves
4.4 Elliptic curve
4.5 Projective plane curve
4.6 Topological classification of compact Riemann surfaces

8th Week (Oct. 21-27)
Midterm

9th Week
5. Sheaf and cohomology
5.1 Sheaf
5.2 Divisor and sheaf
5.3 Operation of sheaves
5.4 Restriction of sheaves and locally free sheaves
5.5 Exact sequence of sheaves

10th Week
5.6 Cech cohomology group
5.7 Ceck cohomology group
5.8 First cohomology group
5.9 Short exact sequences of sheaves and long exact sequence of cohomology groups
5.10 First cohomology groups of various sheaves

11th Week
6. Genus and Riemann-Roch theorem6.1 Genus of a compact Riemann surface
6.2 Riemann-Roch theorem and simple applications

12th Week
6.3 Serre’s duality

13th Week
6.3. Serre’s duality
7. Application of Riemann-Roch theorem
7.1 Topological Euler characteristic and genus
7.2 Genera of various Riemann surfaces

14th Week
7.3 De Rham theorem and Hodge decomposition theorem
7.4 Linear systems and embeddings
7.5 Structure of Riemann surfaces of genera 0 and 1
7.6 Rational maps by divisors of low degrees
7.7 Canonical divisors and their rational maps

15th Week
Reading Period

16th Week (Dec. 16-22)
Final Exam

This course is designed for POSTECH-Yonsei Open Campus Program.
The lectures will be given by Prof. Sung Rak Choi at Yonsei Univ. and Prof. Jihun Park at POSTECH in turn.
The lectures in Yonsei Univ. will be delivered online to POSTECH and the lectures in POSTECH will be given online to Yonsei Univ.

This course is designed for POSTECH-Yonsei Open Campus Program. The lectures will be given by Prof. Sung Rak Choi at Yonsei Univ. and Prof. Jihun Park at POSTECH in turn. The lectures in Yonsei Univ. will be delivered online to POSTECH and the lectures in POSTECH will be given online to Yonsei Univ. at the Zoom meeting room:
Zoom Room
https://yonsei.zoom.us/j/87090341278

Meeting ID: 870 9034 1278