# 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 [6], [7]) 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 [2]).

#### References

 [1] I.R. Shafarevich, "Algebraic surfaces" Proc. Steklov Inst. Math. , 75 (1975) Trudy Mat. Inst. Steklov. , 75 (1975) MR1392959 MR1060325 Zbl 0830.00008 Zbl 0733.14015 Zbl 0832.14026 Zbl 0509.14036 Zbl 0492.14024 Zbl 0379.14006 Zbl 0253.14006 Zbl 0154.21001 [2] V.A. Iskovskikh, "Minimal models of rational surfaces over arbitrary fields" Math. USSR Izv. , 14 : 1 (1980) pp. 17–39 Izv. Akad. Nauk SSSR Ser. Mat. , 43 : 1 (1979) pp. 19–43 MR0525940 Zbl 0427.14011 [3] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001 [4] D. Husemoller, "Classification and embeddings of surfaces" R. Hartshorne (ed.) , Algebraic geometry (Arcata, 1974) , Proc. Symp. Pure Math. , 29 , Amer. Math. Soc. (1975) pp. 329–420 MR0506292 Zbl 0326.14009 [5] R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001 [6] S. Lichtenbaum, "Curves over discrete valuation rings" Amer. J. Math. , 90 : 2 (1968) pp. 380–405 MR0230724 Zbl 0194.22101 [7] I.R. Shafarevich, "Lectures on minimal models and birational transformations of two-dimensional schemes" , Tata Inst. (1966) MR0217068 Zbl 0164.51704

Since 1982 important progress has been made (over the field of complex numbers) in the theory of minimal models for higher-dimensional varieties, and especially for varieties of dimension 3. It has turned out to be necessary to allow a mild type of singularities, namely so-called terminal and canonical singularities. For the precise (very technical) definitions see the references below. (Terminal singularities are special canonical singularities, and for surfaces a point with a terminal (respectively, canonical) singularity is in fact smooth (respectively, a rational double point).) Allowing terminal singularities, the "minimal model problemminimal model problem" (i.e. the existence of a minimal model in a class of birational equivalence) has been solved by S. Mori for varieties of dimension three; in particular, for non-uniruled $3$-dimensional algebraic varieties [a2]. A new phenomenon in the higher-dimensional case is also the non-uniqueness of minimal models. References [a1], [a2] and [a4] are good surveys of this new theory.