Birational Geometry

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:

1. Birational geometry of smooth complex surfaces. (4 lectures)
2. Rational curves — Mori’s Bend-and-Break Method. (4 lectures)
3. Mori’s Cone Theorem. (6 lectures)
4. 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This is the historical reference for the topic.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.