Minimal model program

In algebraic geometry, the minimal model program is part of the birational classification of algebraic varieties. Its goal is to construct a birational model of any complex projective variety which is as simple as possible. The subject has its origins in the classical birational geometry of surfaces studied by the Italian school, and is currently an active research area within algebraic geometry. The basic idea of the theory is to simplify the birational classification of varieties by finding, in each birational equivalence class, a variety which is "as simple as possible". The precise meaning of this phrase has evolved with the development of the subject; originally for surfaces, it meant finding a smooth variety for which any birational morphism with a smooth surface is an isomorphism. In the modern formulation, the goal of the theory is as follows. Suppose we are given a projective variety , which for simplicity is assumed non-singular. There are two cases based on its Kodaira dimension, : We want to find a variety birational to , and a morphism to a projective variety such that with the anticanonical class of a general fibre being ample. Such a morphism is called a Fano fibre space. We want to find birational to , with the canonical class nef. In this case, is a minimal model for . The question of whether the varieties and appearing above are non-singular is an important one. It seems natural to hope that if we start with smooth , then we can always find a minimal model or Fano fibre space inside the category of smooth varieties. However, this is not true, and so it becomes necessary to consider singular varieties also. The singularities that appear are called terminal singularities. Enriques–Kodaira classification Every irreducible complex algebraic curve is birational to a unique smooth projective curve, so the theory for curves is trivial. The case of surfaces was first investigated by the geometers of the Italian school around 1900; the contraction theorem of Guido Castelnuovo essentially describes the process of constructing a minimal model of any surface.