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.
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Outline
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
In the modern formulation, the goal of the theory is as follows. Suppose we are given a projective variety
The question of nonsingularity of the varieties
Minimal models of surfaces
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 Castelnuovo essentially describes the process of constructing a minimal model of any smooth surface. The theorem states that any nontrivial birational morphism f:X→Y must contract a −1-curve to a smooth point, and conversely any such curve can be smoothly contracted. Here a −1-curve is a smooth rational curve C with self-intersection C.C = −1. Any such curve must have K.C=−1 which shows that if the canonical class is nef then the surface has no −1-curves.
Castelnuovo's theorem implies that to construct a minimal model for a smooth surface, we simply contract all the −1-curves on the surface, and the resulting variety Y is either a (unique) minimal model with K nef, or a ruled surface (which is the same as a 2-dimensional Fano fiber space, and is either a projective plane or a ruled surface over a curve). In the second case, the ruled surface birational to X is not unique, though there is a unique one isomorphic to the product of the projective line and a curve.
Higher-dimensional minimal models
In dimensions greater than 2, the theory becomes far more involved. In particular, there exist smooth varieties
The first key result is the cone theorem of Mori, describing the structure of the cone of curves of
The existence of the more general log flips was established by Shokurov in dimensions three and four. This was subsequently generalized to higher dimensions by Birkar, Cascini, Hacon, and McKernan relying on earlier work of Shokurov and Hacon, and McKernan. They also proved several other problems including finite generation of log canonical rings and existence of minimal models for varieties of log general type.
The problem of termination of log flips in higher dimensions remains the subject of active research.