Time and venue: Tuesday and Thursday, 8:30-10 am, MI 00.09.022
Exercise group: S. Neupert, Wednesday, 8:30-10 am, MI 03.08.011.
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 lecture is an introduction to Algebraic Geometry. After a review of some basic theory of algebraic varieties we will introduce the spectrum of a ring, study sheaves and locally ringed spaces, and introduce schemes. We will see examples such as curves and affine and projective spaces. We will also introduce various properties of schemes and tools to study them.
Books on the subject
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
Algebra 1, 2.