Prerequisites
The prerequisites for this course are the standard undergraduate algebra courses on groups, rings and fields (see for example the course notes Algebra 1 and 2 on http://websites.math.
In particular, we assume that the student is familiar with the following notions: group, (commutative) ring, field, homomorphism, ideal, quotient ring, polynomial ring, domain, fraction field, principal ideal domain, unique factorization domain, topological space, continuous map, connected, Hausdorff, compact.
We will use the language of categories when useful, and with the necessary explanations. This material is reviewed during the Mastermath Intensive course on Categories and Modules. This course is now online, see the course page. We recommend going through these videos (and the necessary written material there) during the first weeks of the course.
We will use some results from Commutative Algebra without giving proofs. It is recommended to follow the Commutative Algebra course that is offered on the same day at the same location, but this is not necessary. However, Commutative Algebra is a necessary prerequisite for the follow-up course Algebraic Geometry 2 in spring.
Aim of the course
The course intends to give a first introduction to the basic notions and techniques of algebraic geometry.
We start by introducing the basic objects of algebraic geometry, namely algebraic varieties (affine, projective, general) and the morphisms between them. We then treat basic notions such as dimension, tangent space, differential 1-forms, and smoothness. We will cover applications to curves (including the celebrated Riemann-Roch theorem), and to higher-dimensional algebraic varieties. In particular, we discuss the 27 lines on a smooth cubic surface.
Rules about Homework/Exam
There will be 4 homework sets during the semester, a final written exam and a retake. For the final grade, the homework counts for 20%, and the written exam (or retake) for 80%. To pass the course, the average grade should be at least 5.5, and the grade for the written exam at least 5.0. The homework also counts in the case of a retake.
Homework may be handed in in pairs. In this case you write on the top of your homework with whom you worked together, and you both upload the same solution to the ELO website. Each member of the pair must understand the solutions they hand in.
Lecture Notes/Literature
We will be using the lecture notes Algebraic Geometry by Bas Edixhoven, David Holmes, Martijn Kool, and Lenny Taelman. Corrections to the notes are most welcome.
Video recordings from the academic year 23-24 are available on this page.
Video recordings from the academic year 24-25 are available on this page https://vimeo.com/showcase/11349118 The videos are accessible with password Au0h
Other valuable sources are Harris's Algebraic Geometry: A First Course, Hartshorne's Algebraic Geometry, and Mumford's Red Book of Varieties and Schemes. The book by Harris complements this course particularly well, because it is full of beautiful examples whereas we focus more on general theory.
- Docent: Marton Hablicsek
- Docent: Martijn Kool
- Docent: Amira Tlemsani