Preliminary information

Welcome to the course MAT4215 - Algebraic geometry II. 

The course will give an introduction to scheme theory for the most part following Chapters II and III in Hartshorne. 

Preliminary objectives of the course:

• Sheaves. Understand the key examples and how do computations with them. 
• Schemes. Understand their definition, and why they form the correct generalization of varieties. 
• Geometric properties of schemes and morphisms between them.
• The Proj-construction. Understand the sheaf O(1) and modules on Proj(S).
• Equivalence of line bundles, divisors and invertible sheaves. Maps to projective space. 
• Differentials, the (co)tangent bundle, and the canonical divisor. Emphasis on the case of curves. 
• Sheaf cohomology. Cohomology groups of projective space. Cech cohomology.
• If time permits: Serre duality. Riemann-Roch for curves.