## Class Notes „Algebraic Geometry”

As the syllabus of our Algebraic Geometry class seems to change every couple of years, there are currently three versions of my notes for this class.

### Version of 2021/22

This is the current version of the notes, corresponding to our Algebraic Geometry Master course. It has been updated recently, many errors and inconsistencies in the old versions below have been fixed, and the exposition has been improved significantly in many places. If possible, you should use this new version. It assumes the material of our Commutative Algebra Bachelor class – not very much at the beginning, but more and more so towards the end (so taking both classes in the same semester may be possible). Prior knowledge of our Plane Algebraic Curves Bachelor class is not required, but certainly useful as it gives a more gentle introduction to the field of algebraic geometry, in particular since material specific to (plane) curves has deliberately been left out here in order to avoid significant intersections of the two classes.

- Complete notes (138 pages, last updated December 7, 2022)
- Chapter 0: Introduction
- Chapter 1: Affine Varieties
- Chapter 2: The Zariski Topology
- Chapter 3: The Sheaf of Regular Functions
- Chapter 4: Morphisms
- Chapter 5: Varieties
- Chapter 6: Projective Varieties I: Topology
- Chapter 7: Projective Varieties II: Ringed Spaces
- Chapter 8: Grassmannians
- Chapter 9: Birational Maps and Blowing Up
- Chapter 10: Smooth Varieties
- Chapter 11: The 27 Lines on a Smooth Cubic Surface
- Chapter 12: Schemes
- Chapter 13: Sheaves of Modules
- Chapter 14: Quasi-coherent Sheaves
- Chapter 15: Differentials
- Chapter 16: Cohomology of Sheaves

Any comments or corrections are welcome!

### Version of 2014

This version used to be a Bachelor course some time ago. It can be used as an introduction to algebraic geometry with almost no prerequisites – it connects well with our Commutative Algebra course, but no prior knowledge of this class is assumed. It does not mix very well with our Plane Algebraic Curves class however: the latter did not exist at the time of writing these notes, so there is a substantial amount of intersection.

- Complete notes (133 pages, last updated October 8, 2018)
- Chapter 0: Introduction
- Chapter 1: Affine Varieties
- Chapter 2: The Zariski Topology
- Chapter 3: The Sheaf of Regular Functions
- Chapter 4: Morphisms
- Chapter 5: Varieties
- Chapter 6: Projective Varieties I: Topology
- Chapter 7: Projective Varieties II: Ringed Spaces
- Chapter 8: Grassmannians
- Chapter 9: Birational Maps and Blowing Up
- Chapter 10: Smooth Varieties
- Chapter 11: The 27 Lines on a Smooth Cubic Surface
- Chapter 12: Hilbert Polynomials and Bézout's Theorem
- Chapter 13: Applications of Bézout's Theorem
- Chapter 14: Divisors on Curves
- Chapter 15: Elliptic Curves

### Version of 2002/03

This is the original version of the class notes, which will not be updated any more. However, it covers two semesters, and thus contains more material than the new versions above.

- Complete notes (214 pages, last updated September 28, 2018)
- Chapter 0: Introduction
- Chapter 1: Affine varieties
- Chapter 2: Functions, morphisms, and varieties
- Chapter 3: Projective varieties
- Chapter 4: Dimension
- Chapter 5: Schemes
- Chapter 6: First applications of scheme theory
- Chapter 7: More about sheaves
- Chapter 8: Cohomology of sheaves
- Chapter 9: Intersection theory
- Chapter 10: Chern classes