Lectures on birational geometry

TL;DR

This paper provides comprehensive lecture notes on birational geometry, covering key concepts like minimal models, flips, and the minimal model program, aimed at advanced students and researchers in algebraic geometry.

Contribution

It offers an in-depth, structured overview of birational geometry, including recent developments like flips and finite generation, serving as a valuable educational resource.

Findings

Detailed exposition of the minimal model program

Explanation of flips and their role in birational transformations

Discussion on the existence of minimal models and Mori fiber spaces

Abstract

Lecture notes of a course on birational geometry (taught at College de France, Winter 2011, with the support of Fondation Sciences Math\'ematiques de Paris). Topics covered: introduction into the subject, contractions and extremal rays, pairs and singularities, Kodaira dimension, minimal model program, cone and contraction, vanishing, base point freeness, flips and local finite generation, pl flips and extension theorems, existence of minimal models and Mori fibre spaces, global finite generation, etc.