### Algebraic Geometry 1 - M1 - 8EC

Prerequisites
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 use the language of categories when useful, and with the necessary explanations. This material is reviewed during the Mastermath Intensive course on Categories and Modules. This course is now online, see the course page. We recommend going through these videos (and the necessary written material there) during the first weeks of the course.

We will use some results from Commutative Algebra without giving proofs. It is recommended to follow the Commutative Algebra course that is offered on the same day at the same location, but this is not necessary. However, Commutative Algebra is a necessary prerequisite for the follow-up course Algebraic Geometry 2 in spring.

Aim of the course
The course intends to give a first introduction to the basic notions

and techniques of algebraic geometry.

We start by introducing the basic objects of algebraic geometry,
namely algebraic varieties (affine, projective, general) and the
morphisms between them. We then treat basic notions such as dimension,
tangent space, differential 1-forms, smoothness. We will cover
applications to curves (including the celebrated Riemann-Roch theorem),
and to higher-dimensional algebraic varieties. In particular, we discuss the 27 lines on a smooth cubic surface.

Lecturers