This lecture is intended as an introduction to the modern methods of birational geometry pioneered by Japanese mathematician Shigefumi Mori in the 1980s. It will consist of four parts:
- Birational geometry of smooth complex surfaces. (4 lectures)
- Rational curves — Mori’s Bend-and-Break Method. (4 lectures)
- Mori’s Cone Theorem. (6 lectures)
- An introduction to the Minimal Model Program. (1 lecture)
Requirements: Anyone with a background in complex or algebraic geometry is welcome. For curious students coming from algebra and geometry more broadly, Part 1 is self-motivated and can be attended on its own. Some knowledge about the following objects will be assumed throughout the lecture:
- some culture on complex curves/Riemann surfaces;
- (Part 2-4) basic knowledge on schemes.
- (Part 3-4) basic knowledge on vector bundles/sheaves and cohomology;
Instruction language: English.
Note: I will not use large language models (LLMs) or generative AI to prepare my lectures. In turn, I kindly ask you not to feed any lecture notes, exercise sheets, or other teaching materials that I have provided into LLMs. Thank you.
Commented bibliography:
- Kawamata, Algebraic varieties: Minimal models and finite generation.
This is the reference we will follow most closely. The plan is to cover its Sections 1.1 - 1.7, selected topics from 1.13, and Sections 2.1, 2.3, 2.4, 2.7, and 2.8. The book is quite concise, so we will sporadically complement it with the other references below.
- Debarre, Higher dimensional algebraic geometry.
This book grew out of lecture notes and offers a more gentle introduction to many of the concepts of the lecture. See notably Chapters 1, 2, 3 on rational curves.
- Kollár—Mori, Birational geometry of Algebraic Varieties.
This is the historical reference for the topic. Some of the results presented in this book have since been proven, yet the clear and well-motivated exposition keep it highly relevant. The first chapter gives a particularly nice overview on the Cone Theorem.
- Kursleiter/in: Cécile Gachet