# Algebraic Geometry

In the winter term 2018/19 there will be a course on algebraic geometry. The content is the same as for a usual lecture (4+2SWS). However, as several students were interested in having a seminar on the same subject, the course is now subdivided into:

A seminar (2SWS, Bachelor and Master) taking place twice a week in the first half of the semester. Everyone interested in the subject is invited to attend, also those who do not give a talk.

A lecture (2SWS) taking place twice a week in the second half of the semester, at the same time and place as the seminar.

Exercise sessions for the lecture (2SWS) taking place once per week for the whole semester. In the first half we review the necessary background for the lecture (i.e. do exercises on the material of the seminar), in the second half these are "usual" exercise sessions.

You can participate in the seminar, or in the lecture (including the exercise sessions), or in both.

**Prerequisites:** Algebra 1, 2.

**Time and venue:** We, Fr, 12:15-13:45, 02.08.011

**Exercise group:** S. Liu, Tuesday, 8:30-10:00, 02.04.011.

**Content:** Algebraic Geometry is an intriguing and modern subject with relations to many areas of pure mathematics such as topology, number theory, representation theory, complex geometry, but also to theoretical physics. Classically, algebraic geometry was the study of the geometry of the sets of zeroes of systems of polynomial equations. The field dramatically changed in the 60s and 70s when Grothendieck replaced the previously studied notion of algebraic varieties by his much more general notion of schemes. This new language at first seems to be more technical, and is more abstract. However, it turns out that this approach is in fact more elegant, and at the same time offers much more powerful methods. In the sequel it rapidly became the generally accepted language for this subject, and a rich theory has been developed. This course is an introduction to Algebraic Geometry.

In the seminar we will review of some basic theory of algebraic varieties, introduce the spectrum of a ring, study sheaves and locally ringed spaces, and introduce schemes and some of their basic properties. We will see examples such as curves and affine and projective spaces.

In the lecture we will use these basic notions to study various properties of schemes and tools to study them. In particular, we will study

- dimension of schemes
- products of schemes
- separated and proper morphisms
- coherent sheaves
- differentials and smoothness

**Exercise sheets:** 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14

**References:**

- D. Eisenbud, J. Harris: The Geometry of Schemes
- U. Görtz, T. Wedhorn: Algebraic Geometry I
- R. Hartshorne: Algebraic Geometry
- D. Mumford: The red book of varieties and schemes