**Prerequisites**

The student should have a good working knowledge of varieties over an algebraically closed field, morphisms and rational maps between varieties and Picard Group, for example from the Mastermath Algebraic Geometry 1. Commutative algebra: localisation, exact sequences, integrality, Nakayama's Lemma; the Mastermath course "Commutative Algebra" covers more than enough. Scheme theory, as covered for example in the Mastermath course Algebraic Geometry 2, will be helpful but it is not essential.

**Aim of the course**

The course will introduce the basics of birational geometry for surfaces. We will start with properties of divisors on surfaces, characterization of ampleness and Riemann-Roch for surfaces. We will then focus on the behaviour of rational curves, with the Bend and Break Lemmas and then the Cone Theorem. In the second part of the course, we will describe explicit birational maps between varieties and describe the classification of algebraic surfaces from the point of view of Mori theory. In the end, we will introduce the Minimal Model Program algorithm in higher dimension highlighting the main differences with the two-dimensional case.

**Lecturers**

Marta Pieropan (Utrecht), Diletta Martinelli (UvA)

- Docent: Marta Pieropan