Algebraic Geometry is a very active area of mathematics, linked in nontrivial ways with e.g. commutative algebra, number theory, mathematical physics, and having a number of applications in applied sciences.

In this course we will build the modern dictionary currently shared by all algebraic geometers, starting from the classical world of algebraic varieties (solution sets of polynomial equations), but very soon getting to the language of schemes.

**Tentative Syllabus:**

- Sheaves. Zariski topology on the spectrum of a ring and the definition of schemes.

- Affine and projective schemes. Topological properties, algebraic properties.

- Morphisms of schemes, rational and birational maps, blowups. Normalisation.

- Ring of regular functions, field of rational functions, dimension.

- Fiber products of schemes.

- Properties of morphisms: finite type, separated, proper, projective, flat; infinitesimal properties (smooth, unramified, étale).

- Divisors and line bundles. Picard group.

- Quasicoherent and coherent sheaves. Cohomology.