Introduction to Algebraic Geometry

2017 Enhanced Program for Graduate Study

This is an introduction course to some basic concepts in algebraic geometry, with many examples.  Some commutative algebra is needed. A full knowledge of [AM] will be more than enough for the prerequisite. Students who are interested in a further study are encouraged to take an advanced class covering the materials in [H] or [ShII].

Classroom: 82J12  

Regular meeting time: Monday 2pm-5pm (with some classes shuffled)

Email: cyxu [ at ] math.pku.edu.cn

Textbook: Basic Algebraic Geometry I (Igor Shafarevich)

Tentative Schedule

1. Feb. 20  Chapter I.1  Plane Curves

2. Feb. 25  Chapter I.2 and 3  Closed subset and Rational function

3. Feb. 27  Chapter I.4 Quasi-projective varieties

4. Mar. 6 Chapter I.5 Products and maps of quasi-projective varieties  Pset 1

5. Mar. 12 Chapter I.6 Dimension

6. Mar. 20 Chapter II.1 Singular and smooth points

7. Mar. 25 Chapter II.2 Power series expansion

8. Mar. 27 Chapter II. 3 and 4  Properties of smooth points and birational maps

9.  Apr. 10 Middle term exam  (Apr. 3 Lectures on exercises Pset 2)

10. Apr. 15 Chapter II. 5 Normal varieties

11. Apr. 17 Chapter II.6 Singularities of a map

12. Apr. 21 Chapter III.1 Divisors

13. May 4 Chapter III.2 and 3 Divisors on curves and plane cubics

14. May 8 Chapter III.4 and 5 Algebraic groups and Differential forms

15. May 22 Chapter III.6 Examples and applications

16. June 17 9am-12pm Final Exam

Homework: 

Pset 1: Ex 1.1, Problem 3, 5, 6, 10.  Ex 1.2, Problem 1, 5, 13, 19.  Ex 1.3, Problem 4, 7.  Ex 1.4, Problem 1, 3, 5, 6. 

Pset 2: Ex 1.5, Problem 1, 4, 7, 9.  Ex 1.6,  Problem 2, 4, 6, 7, 8.  Ex 2.1, Problem 5, 6, 9, 12, 13, 15.  Ex 2.1, Problem 3, 5, 6. Ex 2.1 1, 3. Ex 2.3, Problem 1, 4, 6, 8, 11. Ex 2.4, 4, 9, 10.

P set 3: Ex 2.5, Problem 1, 2, 5, 6. Ex 2.6, 1-6.

Other literature: 

[AM] Introduction to Commutative Algebra (Atiyah-Macdonald)

[H] Algebraic Geometry (Hartshorne)

[ShII] Basic Algebraic Geometry II (Igor Shafarevich)