We start by introducing the basic objects of algebraic geometry, namely varieties and the morphisms between them. We then treat basic notions such as dimension, tangent space, differential 1-forms, smoothness. Towards the end we will focus on curves and surfaces, and we hope to accomplish a proof of the Riemann hypothesis for curves over finite fields. This result allows one to give good estimates for the number of points on such curves. These estimates are essential in many applications in coding theory and cryptography.

The prerequisites for this course are the standard undergraduate algebra courses on groups, rings and fields (see for example the course notes Algebra 1 and 2 on http://websites.math.leidenuniv.nl/algebra/ or chapter I-IV of S. Lang's 'Algebra', Springer GTM 211)

In particular, we assume that the student is familiar with the following notions: group, (commutative) ring, field, homomorphism, ideal, quotient ring, polynomial ring, domain, fraction field, principal ideal domain, unique factorization domain, topological space, continuous map, connected, Hausdorff, compact.

We will occasionally use the language of categories. This material is reviewed during the Mastermath "Intensive course on Categories and Modules" that we strongly recommend taking in the first week of the Fall semester.