Minimal model (birational geometry)

In algebraic geometry, more specifically in the field of birational geometry, the theory of minimal models is part of the birational classification of algebraic varieties. Its goal is to construct, given a variety satisying certain restrictions, a birational model of that variety which is, in some sense, as simple as possible. The subject has its origins in the classical birational geometry of the Italian school, and is currently an extremely active research area within algebraic geometry.

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, it meant finding a smooth variety $X$ for which any birational morphism $f:\; X\text{'}\; ightarrow\; X$ is an isomorphism.

In the modern formulation, the goal of the theory is as follows. Suppose we are given a projective variety $X$, which for simplicity is assumed nonsingular. There are two cases:
*If $X$ has Kodaira dimension $kappa(X,K\_X)=-1$, we want to find a variety $X^prime$ birational to $X$, and a morphism $f:\; X\text{'}\; ightarrow\; Y$ to a projective variety $Y$ such that dim $Y$< dim$X\text{'}$, with the anticanonical class $-K\_F$ of a general fibre $F$ being ample. Such a morphism is called a "Fano fibre space".
* If $kappa(X,K\_X)$ is at least 0, we want to find $X\text{'}$ birational to $X$, with the canonical class $K\_\{X^prime\}$ nef. In this case, $X\text{'}$ is a "minimal model" for $X$.

The question of nonsingularity of the varieties $X\text{'}$ and $X$ appearing above is an important one. It seems natural to hope that if we start with smooth $X$, then we can always find a minimal model or Fano fibre space inside the category of smooth varieties. However, it turns out that this is not true, and so it becomes necessary to consider singular varieties also. The issue of which class of singularities are to be incorporated in the theory is a subtle and technically difficult one.

Minimal models of surfaces

Every irreducible complex algebraic curve is birational to a unique smooth projective curve, so there is no meaningful theory for curves. As mentioned above, 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 $f:\; Y\; ightarrow\; X$ must contract a $(-1)$-curve to a smooth point, and conversely any such curve can be smoothly contracted. Here by a "(-1)-curve" we mean a smooth rational curve $C$ with self-intersection $C^2=-1$. Note that any such curve must have $K\_Y\; cdot\; C=-1$; this indicates the link between minimality and nefness of the canonical class.

Castelnuovo's theorem says that to construct a minimal model for a smooth surface, we simply contract all the $(-1)$-curves on the surface, and the resulting variety $X\text{'}$ is either a minimal model, or a Mori fibre space. In fact, more was proved: the only possibilities for $X\text{'}$ are the following:

1. $K\_\{X\text{'}\}$ is nef;

2. $X\text{'}$ is a ruled surface over a curve $C$;

3. $X\text{'}$ is isomorphic to the projective plane $mathbf\{P\}^2$.

For a given variety $X$, the end product $X\text{'}$ of this process may not be unique. However, it turns out that, in the first case above, $X\text{'}$ is unique.

Higher-dimensional minimal models

In dimensions greater than 2, the theory becomes far more involved. In particular, there exist smooth varieties $X$ which are not birational to any smooth variety $X\text{'}$ with nef canonical class. The major conceptual advance of the 1970s and early 1980s was that the construction of minimal models is still feasible, provided one is careful about the types of singularities which occur. (For example, we want to decide if $K\_\{X\text{'}\}$ is nef, so intersection numbers $K\_\{X\text{'}\}\; cdot\; C$ must be defined. Hence, at the very least, our varieties must have $nK\_\{X\text{'}\}$ Cartier for some positive integer $n$.)

The first key result is the Cone theorem of Mori, describing the structure of the cone of curves of $X$. Briefly, the theorem shows that starting with $X$, one can inductively construct a sequence of varieties $X\_i$, each of which is 'closer' than the previous one to having $K\_\{X\_i\}$ nef. However, the process may encounter difficulties: at some point the variety $X\_i$ may become 'too singular'. The conjectural solution to this problem is the flip, a kind of codimension-2 surgery operation on $X\_i$. It is not clear that the required flips exist, nor that they always terminate (that is, that one reaches a minimal model $X\text{'}$ in finitely many steps.) Mori showed in 1988 that flips exist in the 3-dimensional case; much recent work has focused on existence and termination problems in higher dimensions.

References

* Mori, S. "Flip theorem and the existence of minimal models for 3-folds", J. Amer. Math. Soc. 1 (1988), no. 1, 117--253.
* Kollár, J. and Mori, S., "Birational Geometry of Algebraic Varieties", Cambridge University Press, 1998. ISBN 0-521-63277-3