## Algebraic Geometry

### Prof. Barbara Fantechi

- Algebraic varieties: definition, affine and projective varieties,
examples. Products, fibered products. Separatedness and completeness.
Irreducibility and dimension theory. Tangent spaces and differentials.
Smoothness.
- Schemes: definition, affine and projective schemes, examples. Reduced
schemes and varieties. Products and fibered products. Coherent,
locally free, invertible sheaves. Cotangent sheaf.
- Smooth projective curves, correspondence
divisors/invertible sheaves, Serre duality, Riemann-Roch,
applications.