## Algebraic Geometry

### Prof. Barbara Fantechi

Content:

- Schemes. We will cover the material of chapters 2 and 3 of Hartshorne's book; in particular: schemes, quasi-coherent sheaves, projective morphisms, tangent and cotangent sheaves, sheaf cohomology, flatness, smoothness, properties of flat morphisms (in particular cohomology and ba se change theorem).
- Moduli spaces problems and the need for algebraic stacks: Deligne-Mumford and Artin algebraic stacks, extensions of properties of schemes to algebraic stacks (fiber products, representability, properness, separatedness). The stress will be on examples and on how to work with algebraic stacks.