**[Fall 2020]**

**Prerequisites**

Varieties over an algebraically closed field (for example Mastermath Algebraic Geometry 1). Algebra including rings, modules and Galois theory (for example, chapters I-VI of Lang's "Algebra", or see the course notes for Algebra 1, 2, 3 available here: http://websites.math.leidenuniv.nl/algebra/ ). Commutative algebra: localisation, exact sequences, integrality, Nakayama's Lemma; the Mastermath course "Commutative Algebra" covers more than enough. We will also freely make use of material covered in the Mastermath intensive course on categories and modules.

**Aim of the course**

This course will introduce the study of rational points on higher-dimensional varieties, concentrating on surfaces. On the geometric side, we will cover the basic geometry of surfaces including divisors, the Picard group, intersection theory and the Riemann-Roch theorem; on the algebraic side, we will introduce p-adic numbers, the Hasse principle and Brauer groups of fields. The two branches come together with the Brauer-Manin obstruction.

**Lecturers**

Martin Bright (Leiden), Ronald van Luijk (Leiden)

- Docent: Martin Bright
- Docent: Ronald van Luijk