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