• Corpus ID: 237353527

Effective results in the three-dimensional minimal model program

@inproceedings{Prokhorov2021EffectiveRI,
  title={Effective results in the three-dimensional minimal model program},
  author={Yuri Prokhorov},
  year={2021}
}

• Yuri Prokhorov
• Published 30 August 2021
• Mathematics

We give a brief review on recent developments in the three-dimensional minimal model program. In this note we give a brief review on recent developments in the three-dimensional minimal model program (MMP for short). Certainly, this is not a complete survey of all advances in this area. For example, we do not discuss the minimal models of varieties of non-negative Kodaira dimension, as well as, applications to birational geometry and moduli spaces. The aim of the MMP is to find a good representative in a fixed birational equivalence class. Starting with an arbitrary smooth projective variety one can perform a finite number of certain elementary transformations, called divisorial contractions and flips, and at the end obtain a variety which is simpler in some sense. Most parts of the MMP are completed in arbitrary dimension. One of the basic remaining problems is the following: Describe all the intermediate steps and the outcome of the MMP. The MMP makes sense only in dimensions ≥ 2 and for surfaces it is classical and well-known. So the first non-trivial case is the three-dimensional one. It turns out that to proceed with the MMP in dimension ≥ 3 one has to work with varieties admitting certain types of very mild, socalled terminal, singularities. On the other hand, dimension 3 is the last dimension where one can expect effective results: in higher dimensions classification results become very complicated and unreasonably long. We will work over the field C of complex numbers throughout. A variety is either an algebraic variety or a reduced complex space.