Lectures on birational geometry

Birational geometry, that is, the classication of algebraic varieties up to birational equivalence occupies a central position in algebraic geometry. The machinery for carrying out birational geometry is the so‑called minimal model program (MMP for short). The MMP is a specific process designed to identify nice elements in a birational class and then to classify them. These nice elements are of two very different types: minimal models and Mori fibre spaces. Minimal models have a certain positivity property and Mori fibre spaces have a certain negativity property. Curves of genus at least one and Calabi‑Yau varieties are examples of the first type but a Fano variety is an example of the second type. Some of the general ideas for finding minimal models and Mori fibre spaces go back to the early 20^th century but it was in the 1970's and afterward that modern points of view emerged which accumulated in the proof of some of the main conjectures of the field in recent years.

Birational geometry is primarily concerned with smooth projective varieties (over the complex numbers for much of this course). However, to apply the MMP we first have to deal with singularities as they appear in the study of extremal rays and contractions. Second, we need to treat flipping contractions and prove that their corresponding flips exist. Third, we have the problem of proving that sequences of flips terminate. Fourth comes the study of pluricanonical systems on minimal models and the abundance problem. In addition, there are several other problems concerning the behavior of singularities and Fano varieties.

In this course, by starting from scratch I will try to give a detailed account of many of the main tools, methods, and fundamental results in the subject including the progress made in recent years, and of what remains to be solved. Here is a list of some of the topics that I hope to cover in the course: birational geometry of curves and surfaces, rational curves, adjunction, pairs and the various classes of singularities, Kodaira‑type vanishing theorems, extension theorems, extremal rays, minimal models, Mori fibre spaces, nonvanishing, base point freeness, the cone and contraction, flips, termination, MMP, Shokurov algebras, finite generation of canonical rings, ACC conjectures on singularities, Sarkisov program, rationally connected varieties, Iitaka conjecture, and abundance.

In algebraic geometry, one usually has to learn a lot of material before feeling a sense of direction. A pedagogical advantage of birational geometry is that one can see advanced techniques of algebraic geometry applied to concrete geometric problems.