Welcome to the course "Algebraic Geometry" in the winter term 2024/2025!
General information
What this course is about
Algebraic geometry is the study of zero sets of (systems of) polynomials.
Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. As a study of systems of polynomial equations in several variables, the subject of algebraic geometry begins with finding specific solutions via equation solving, and then proceeds to understand the intrinsic properties of the totality of solutions of a system of equations. This understanding requires both conceptual theory and computational technique. — Wikipedia
What you will learn
You will learn about the interplay between algebra and geometry. You will develop geometric intuition. You will learn basic notions like affine and projective varieties and their morphisms; properties like dimension, smoothness, and degree; classical constructions like the Veronese map and the Grassmannian; examples like rational normal curves and Segre varieties; you will get a glimpse of the birational classification which is still a major research area.
What we will do
We will follow a nice little book titled An Invitation to Algebraic Geometry by K. Smith et. al. (2000). This is available for free via our library. The plan is to cover the whole book. I have (hand-)written up my own version—with more colors but basically the exact same content. You can find it in my notes repository (filter for this course), and linked in consecutive order below. The password was/will be given in class.
Formalities
This module is worth 9 CP. More information is in the module handbook. The lectures are on Tuesdays 10:00–11:30 and Thursdays 10:00–11:30 in 48-438, starting on October 22, 2024. The last lecture is on February 6. The winter break is from December 23 to January 3. Additionally, we have exercise sessions on Tuesdays 15:30–17:00 in 48–438, starting from October 22, 2024. This semester I will take care of the exercise sessions by myself. An exercise sheet will be released each Tuesday on my website.
For the exam a thorough understanding of all the lecture material below (including the exercises) is essential.
For experts
For those who have basic knowledge of algebraic geometry already or who need more material per time, I recommend you read the book Algebraic Geometry. A first course by J. Harris (1992) in parallel! This book is available for free via our library as well.
Lecture material
- Affine algebraic varieties
- The algebra-geometry dictionary
- Projective varieties
- Quasi-projective varieties
- Classical constructions in algebraic geometry
- Smoothness in algebraic geometry
- Birational geometry
Further material
- Blowup in a point (Beautiful animation of a blowup, low resolution unfortunately.)