We are glad to welcome you to Algebraic Geometry. This course is an introduction to the world of algebraic geometry. We will start by introducing affine varieties, which are zero loci of polynomials in affine space, and study their topological properties. With the help of the language of sheaves, mimicking what happens in differential geometry, we will glue varieties into more general (abstract) varieties. In particular, we will study projective varieties and the construction of new varieties using products, blow-ups and Grassmannians. You will learn about topological properties such as irreducibility, dimension, complete and separated varieties and local properties, such as smoothness. Toward the end of the course we will introduce the language of schemes, which completely revolutionized the field in the 60's, when it was introduced by Grothendieck. A key tool for studying the geometric property of schemes are quasi-coherent sheaves, such as the twisting sheaves O(n), skyscraper sheaves, and sheaves of differentials. We will learn how these are constructed and how to compute their cohomology.

You will perhaps need to register to maintain access to this course in Canvas after the web registration period.

Course content

Textbook: [G] Andreas Gathmann, Algebraic Geometry, version of August 5th, 2024, downloadable at this link.

We are roughly going to cover one chapter per lecture. Here is a preliminary outline:

Lecture 1. Affine varieties and their topology
Lecture 2. Dimension theory
Lecture 3. The sheaf of regular functions
Lecture 4. Morphisms
Lecture 5. Varieties
Lecture 6. Projective varieties I: topology
Lecture 7. Projective varieties II: ringed spaces
Lecture 8. Grassmannians
Lecture 9. Birational maps and blowing up
Lecture 10. Smooth varieties
Lecture 11. Affine schemes
Lecture 12. Schemes
Lecture 13. Sheaves of modules
Lecture 14. Quasi-coherent sheaves
Lecture 15. Differentials
Lecture 16. Cohomology of sheaves

Note that this outline and the precise contents most likely will change during the course. For detailed information about each lecture, see Modules. For the schedule, look up the course (SF2716) in TimeEdit, the schedule in your KTH menu, or this pdf (as of Jan 9).

If you want to learn more about algebraic geometry beyond this course, you could start by reading one of the following books on schemes:

Ellingsrud–Ottem, Introduction to Schemes, Dec 2024, web page.
Hartshorne, Algebraic Geometry, 1977, SpringerLink.
Görtz–Wedhorn, Algebraic Geometry I: Schemes, 2020, 2nd edition, SpringerLink.
Mumford, The Red Book of Varieties and Schemes, 1999, SpringerLink.

Examination

The course is examined through written assignments consisting of 4 problem sets to be solved and handed in during the course and a final oral exam.

Assignment 1 (due Feb 7)
Assignment 2 (due Mar 14)
Assignment 3 (due Apr 11)
Assignment 4 (to appear, due May 16)

The homework should be handed in as a pdf online, see Assignments. To be admitted to the oral exam, you need to score at least 50% on the written assignments. To pass the course, you need to pass the written assignments and the oral exam.

Grades

The written assignments and the oral exam will be scored from 0–100 where passing is 50 or above (preliminary). The final grade will depend on the unweighted mean of the two scores with the following preliminary scale:

Grade	A	B	C	D	E
Mean score	90+	80-89.9	70-79.9	60-69.9	50-59.9

Contact information

You can find all announcements posted via Announcements. In this course, you will meet the following teachers:

David Rydh (examiner and lecturer): dary@math.kth.se
Sofia Tirabassi (lecturer): tirabassi@math.su.se
David Kern (teaching assistant): dkern@kth.se

Under the People tab, you can find the participants of this course.