Vorlesung: Algebraische Geometrie

Algebraic Geometry is an exciting and very active research area in mathematics. Its starting point is the study of systems of polynomial equations in several variables. Algebraic geometry has connections with many topics in mathematics, e.g., complex goemetry, number theory, topology, but also to theoretical physics.

The main focus of this lecture course will be on the introduction of Grothendieck’s language of schemes. The notion of scheme generalizes the classical notion of variety and offers a more elegant approach to many questions. The introduction of schemes in the 60’s and 70’s has revolutionized algebraic geometry; nowadays this is the universally accepted foundation of algebraic geometry.

The higher level of generality, on the other hand, requires an additional effort. As we will see in the course, this pays off in the long run. The full power of Grothendieck’s approach will become visible when studying advanced topics, for which this lecture course hopefully can prepare the grounds.

Dates: Lectures: Tue 10am-12pm, Wed 10am-12pm; Exercise group: Wed, 12-2pm. Starting Oct. 25: Tue, 12-2pm. The exercise group will start on Oct. 19.

Place: Lectures: S-U-3.01. Exercise group: N-U-4.01.

Prerequisites: Good knowledge in linear algebra in algebra, and in commutative algebra (as covered by my course Algebra II , say).

Contents of the course: Here you find information on the contents of the Algebraic Geometry 1 course I taught in the winter term 2018/19 and remarks about what would be useful preparation. This course will be similar.

Literature

Survey: To get some overview, you can take a look at my survey paper Classics revisited: EGA

There are plenty of text books on algebraic geometry. I will list a small selection here. As usual, it is recommended that you take a look at several of these books (many can be found online) before you would buy one.

With Torsten Wedhorn, I have written a book on algebraic geometry which I can recommend :-)

  • Görtz, Wedhorn, Algebraic Geometry I. Schemes. With Examples and Exercises. Vieweg.

This book contains basically everything that will be covered in the course, and a lot more.

A good introduction which covers many essential topics is Mumford’s classic:

A relatively new book (project) is

which covers the typical content of the lecture classes Algebraic Geometry 1 and 2 (and quite a bite more), and where a lot of attention is paid on motivating things thoroughly.

Further standard references are

  • Dieudonné, J., Grothendieck, A., Eléménts de géométrie algébrique I-IV, Publ. Math. IHES 8, 11, 17, 20, 24, 28, 32 (1960-67), gescannt auf numdam.org
  • Hartshorne, R., Algebraic Geometry, Springer Graduate Texts in Mathematics 52
  • Shafarevich, I., Basic Algebraic Geometry 1, 2, Springer

Hartshorne’s book is much more comprehensive than Mumford’s, in particular if you also count the material covered in the exercises. Since quite a few of the exercises are quoted in the text, the book is not an easy read. EGA is the original, and the standard reference for the foundations of Grothendieck’s algebraic geometry. All proofs are spelled out in detail, and the authors strive for the greatest possible generality. As a result, these volumes have a lot of pages. Nevertheless, at some point (but maybe only when you continue studying algebraic geometry beyond this course) it is certainly worth the time, to take a look and to get a rough overview of what is covered where. It is written in French (as many important texts in algebraic geometry, so learning the basics of “mathematical French” is advisable if you want to specialize in this area). The Stacks project which was initiated by A. J. de Jong by now has outgrown even the size of EGA, by far, and contains many more recent results. The books by Shafarevich are a good complement to the other ones.

A more recent book which is more accessible than EGA, but in many parts more general than Hartshorne, is

  • Liu, Q., Algebraic Geometry and Arithmetic Curves, Oxford Univ. Press

There are numerous other books on algebraic geometry, for instance

  • Bosch, S., Algebraic Geometry and Commutative Algebra, Springer
  • Harder, G., Lectures on Algebraic Geometry I, II, Vieweg-Teubner

The following two books (among numerous others) deal with algebraic geometry specifically over the complex numbers, and its relation to complex geometry:

  • Griffiths, P., Harris, J., Principles of Algebraic Geometry, Wiley Interscience
  • Huybrechts, D., Complex geometry, Springer Universitext

Finally, there are many lecture notes (of differing quality) on algebraic geometry to be found in the net. For instance the following can be recommended.

In Milne’s script, there are further pointers to the literature (with comments that I mostly agree with). A more comprehensive list of lecture notes was compiled by Franz Lemmermeyer.