# Minimal model

An algebraic variety which is minimal relative to the existence of birational morphisms into non-singular varieties. More precisely, let $B$ be the class of all birationally-equivalent non-singular varieties over an algebraically closed field $k$, the fields of functions of which are isomorphic to a given finitely-generated extension $K$ over $k$. The varieties in the class $B$ are called projective models of this class, or projective models of the field $K/k$. A variety $X\in B$ is called a relatively minimal model if every birational morphism $f\colon X\to X_1$, where $X_1\in B$, is an isomorphism. In other words, a relatively minimal model is a minimal element in $B$ with respect to the partial order defined by the following domination relation: $X_1$ dominates $X_2$ if there exists a birational morphism $h\colon X_1\to X_2$. If a relatively minimal model is unique in $B$, then it is called the minimal model.

In each class of birationally-equivalent curves there is a unique (up to an isomorphism) non-singular projective curve. So each non-singular projective curve is a minimal model. In the general case, if $B$ is not empty, then it contains at least one relatively minimal model. The non-emptiness of $B$ is known (thanks to theorems about resolution of singularities) for varieties of arbitrary dimension in characteristic 0 for and for varieties of dimension $n\leq3$ in characteristic $p>5$.

The basic results on minimal models of algebraic surfaces are included in the following.

1) A non-singular projective surface $X$ is a relatively minimal model if and only if it does not contain exceptional curves of the first kind (see Exceptional subvariety).

2) Every non-singular complete surface has a birational morphism onto a relatively minimal model.

3) In each non-empty class $B$ of birationally-equivalent surfaces, except for the classes of rational and ruled surfaces, there is a (moreover, unique) minimal model.

4) If $B$ is the class of ruled surfaces (cf. Ruled surface) with a curve $C$ of genus $g>0$ as base, then all relatively minimal models in $B$ are exhausted by the geometric ruled surfaces $\pi\colon X\to C$.

5) If $B$ is the class of rational surfaces, then all relatively minimal models in $B$ are exhausted by the projective plane $P^2$ and the series of minimal rational ruled surfaces $F_n=P(\mathcal O_{P^1}+\mathcal O_{P^1}(n))$ for all integers $n\geq2$ and $n=0$.

There is (see , ) a generalization of the theory of minimal models of surfaces to regular two-dimensional schemes. Minimal models of rational surfaces over an arbitrary field have been described (see ).

How to Cite This Entry:
Minimal model. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Minimal_model&oldid=43462
This article was adapted from an original article by V.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article