**Introduction to algebraic geometry**

**by Prof. Ugo Bruzzo**

**1. Complex manifolds.** Basic definitions and properties. Examples.
Holomorphic vector bundles.

**2. Sheaf cohomology.** Basic homological algebra.
De Rham cohomology. Presheaves and sheaves. Cech cohomology.
Cup product. Chern classes of vector bundles.

**3. Divisors.** Divisors, meromorphic functions and line bundles on
Riemann surfaces. The higher dimensional case. Bertini's theorem. The
adjunction formula.

**4. Algebraic curves.** Serre duality. The Kodaira embedding.
Branched coverings. The Riemann-Hurwitz formula. The *g*=0 and *g*=1 cases.
The Weierstrass representation of elliptic curves. Jacobian varieties.
Blow-up. Nodal curves.

**5. Algebraically integrable systems. ** Spectral curves. Flow
linearization on the Jacobian variety. Examples.