Course Identification

1. First course in ring theory (you should know what a UFD is and the definition of a module).
2. Second course in topology (you should know how two dimensional manifolds are classified by the Euler characteristic).
3. First course in complex analysis (up to and including the residue formula).
4. Real analysis in several variables (you should know the definition of a manifold and what a partition of unity is).

Algebraic Geometry, mostly over the complex numbers. There will be two parts for the course and they will be given in parallel.

The first part will be about algebraic curves. Topics I plan to cover are:

1. Affine and projective plane curves.
2. Bezout’s theorem.
3. Hurwitz’s theorem, the degree—genus formula.
4. Differentials on curves.
5. Abel’s theorem, Jacobian.
6. Riemann—Roch.

The second part will be about higher dimensional varieties. Topics I plan to cover are:

1. Algebraic sets, Hilbert’s theorem on zeros.
2. The category of affine varieties. Chevalley’s theorem.
3. Finite maps.
4. Dimension theory.
5. Line bundles.
6. Sheaves and cohomology.

Upon successful completion of the course, the students will be able to:

• Translate geometric statements to algebraic ones and vice versa.
• Know and use the principal techniques of algebraic geometry: generic point method, preservation of number principle, and fibering over a lower dimensional variety.
• Compute cohomologies of line bundles on curves in simple cases.
• Take a scheme-based Algebraic Geometry course.

I will borrow heavily from Shafarevich’s “Basic Algebraic Geometry” and Kirwan’s “Complex Algebraic Curves”.