I. Contents：
This course is intended to introduce the modern language in algebraic geometry. Classically, the main objects in algebraic geometry are algebraic varieties. However, it turns out that it’s more convenient to consider the setting of schemes which involve richer structure. The following is the plan:
1. Varieties (affine and projective)
2. Why scheme?
3. Sheaves
4. Definition of schemes
5. Properties of schemes
6. Divisors
7. Projective morphisms
8. Differentials
9. Formal scheme
10. Sheaf cohomology
11. Cohomology of projective spaces
12. Serre duality
II. Course prerequisite：
Algebra is required. Some experience in commutative algebra is preferred..
III. Reference material ( textbook(s) )：1. R. Hartshorne: Algebaic Geoemtry, Graduate Text in Mathematics 52, Springer Verlag
2. D. Eisenbud; J. Harris: The Geometry of Schemes, Graduate Text in Mathematics, Springer Verlag
IV. Grading scheme：
Homework 50%
Presentation 30%
Participation 20%