Lectures on birational geometry (séance n°16) -
par Caucher Birkar, lauréat 2010 du Prix de la Fondation Sciences mathématiques de Paris
- Cours de la Fondation Sciences mathématiques de Paris
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 di fferent 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 fi rst 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 20th 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.
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