# Algebraic geometry 1

**Schedule **for the course:

Monday 10-12 Uhr WSC-N-U-4.05

Wednesday 10-12 Uhr WSC-N-U-4.05

**Exercise classes:** Friday 12-14 WSC-N-U-4.05

There is a moodle-page with exercises, information on the content of the course and room for dicussions (AlgebraischeGeometrie helps to sign up for the course).

**Content:** The course aims to give an introduction to algebraic geometry - classically this meant the study of the geometry of the set of solutions of a system of polynomial equations. You might have seen in a course on commutative algebra, that the set of solutions can be encoded efficiently in terms of rings and their prime ideals.

This language has the advantage that once one manages to translate geometric notions like dimension, smoothness, tangent spaces in these terms one can apply these even to study solutions of equations over rational numbers, integers or finite fields, which is quite unexpected and very useful.

As for manifolds, once one has an understanding of basic geometric objects, it is useful to allow to patch these together. This leads to the concept of schemes.

In the course we will learn about schemes and their geometric properties and will then apply these to obtain some classical results on the geometry of curves.

**Prerequisites:** Algebra and some commutative Algebra: Rings, modules, ideals, prime ideals, localization, tensor products. If you are willing to accept results of commutative algebra without proof these notions will be sufficient - and even a little less might be ok.

**Literature: **There is a wealth of good books on algebraic geometry, have a look at a few of them before deciding on one. Here is a small selection:

More recent books:

R. Vakil: Foundations of Algebraic Geometry (Notes) http://math.stanford.edu/~vakil/216blog/

This is a recent book project. I will try to use it as a rough guideline.

U.Görtz, T. Wedhorn: Algebraic geometry I, Schemes with Examples and Exercises, Vieweg.

Here you will find much more about the basics on schemes than one could cover in a semester. It is an extremely useful reference.

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

Older classics:

R. Hartshorne: Algebraic Geometry, Springer Graduate Texts in Mathematics 52.

Many mathematicians learned algebraic geometry from this book. The ratio content / length is still very high. The exercises are an essential part of this book.

D. Mumford: The Red Book on Varieties and schemes. Springer Lecture Notes in Math. 1358

I. Shafarevich: Basic Algebraic Geometry 1, 2, Springer

J. Dieudonné, A. Grothendieck: Eléménts de géométrie algébrique I-IV, Publ. Math. IHES 8, 11, 17, 20, 24, 28, 32 (1960-67), gescannt auf numdam.org

The classic reference for the last (up to now) revolution in algebraic geometry. It is still the standard reference - but it was probably never intended to be a book to use for a course.

A recent resource is the Stacks-Project of A.J. deJong.

This is an amazing reference, containing much more than any series of lecture courses could cover. It is hyperlinked and searchable so that you can look for particular results and their proofs very easily.