This is a rather sketchy (yet hopefully motivated) introduction to minimal models and the Enriques classification of complex algebraic surfaces. In the sequel, a surface means a smooth projective variety of dimension 2 over . Our main source is . See also  and .
Let be a surface. The structure of rational maps between and any projective variety is simple by the following
The structure of birational morphisms between surfaces is also rather simple. Suppose is a birational morphism of surfaces. Then can be decomposed as a sequence of blow-ups , and an isomorphism . Combining this fact with the elimination of indeterminacy, we obtain that
The exponential exact sequence induces a long exact sequence Hodge theory shows that is a lattice in , so the Picard variety is a complex torus (and indeed an abelian variety). Since , we obtain an exact sequence where is the image of .
Since is finitely generated, the Neron-Severi group is also finitely generated. The rank of is called the Picard number of . The Neron-Severi group can also be described as the group of divisors on modulo algebraic equivalence ([3, V 1.7]).
The behavior of the Neron-Severi group under a blow-up can be easily seen.
In other words, a blow-up increases the Picard number by 1. Thus the Neron-Severi group gives us a canonical order on the set of isomorphism classes of surfaces birationally equivalent to a given surface .
In particular, every surface dominates a minimal surface since the Picard number is finite. Conversely, every surface is obtained by a sequence of blow-ups of a minimal surface. So the problem of birational classification boils down to classifying minimal surfaces.
We can characterize minimal surfaces as those without exceptional curves. By the very definition of blow-ups, an exceptional curve is isomorphic to and has self-intersection number . This is actually a useful numerical criterion of minimality of surfaces ([1, II. 17]).
In most cases, has a unique minimal element. However, the situation is not that simple for a large class of surfaces — ruled surfaces.
As a little digression, we can calculate several birational invariants for ruled surfaces easily.
Now let us step back to the problem of finding the minimal models of ruled surfaces. This is closely related to the notion of geometrically ruled surfaces.
The first thing to notice is that geometrically ruled surfaces form a subclass of ruled surfaces due to the following theorem ([1, III.4]).
Also from the Noether-Enriques theorem, we know that every geometrically ruled surface over admits local trivializations as a -bundle, thus they are are classified by the cohomology group . The exact sequence gives a long exact sequence Since , classifies rank 2 vector bundle over and , one knows that every geometrically ruled surface over is actually -isomorphic to for some rank two vector bundle over . Moreover, such and are -isomorphic if and only if for some line bundle over .
Now let us see how geometrically ruled surfaces play a significant role in classifying minimal ruled surfaces.
In other words, for an irrational ruled surface, its minimal models are not unique and are classified by those projective bundles : the theory of rank two vector bundles over a curve is delicate, but more or less understood.
The ruled surfaces with base curve are called Hirzebruch surfaces. In particular, they are rational surfaces. Among these, the only geometrically ruled ones are () since every vector bundle over is a direct sum of line bundles. The above classification of minimal models of ruled surfaces fails for Hirzebruch surfaces. A calculation of intersection numbers on ruled surface implies the following result ([1, IV.1]).
It follows that 's are distinct and minimal for . However, it also follows that there is an irreducible curve on with . Hence by the uniqueness, coincides with the blow-up of at one point, hence is not minimal.
In order to find minimal models for rational surfaces, we need the following nontrivial fact ([1, V.6]). Notice that any rational surface satisfies ().
Now we can deduce the following classification of minimal models of rational surfaces.
A similar argument using above useful lemma implies the following numerical characterization of rational surfaces ([1, V.1]).
Castelnuovo's Rationality Criterion together with the usage of Albanese varieties will enable us to finally show the uniqueness of the minimal models of all non-ruled surfaces ([1, V.19]).
Hence we have found a complete list of minimal surfaces by now.
Castelnuovo's Rationality Criterion provides a handy numerical tool to distinguish rational surfaces from others. We would like to see how the birational invariants will help us classifying surfaces. This can be achieved for ruled surfaces as well ([1, VI.18]).
In view of the important role played by the plurigenera, we introduce the notion of Kodaira dimension for a smooth projective variety.
In particular, the Kodaira dimension is always no greater than the dimension of . Some examples are in order.
From this point of view, Kodaira dimension surfaces are analogous to rational curves, Kodaira dimension 0,1 surfaces are analogous to elliptic curves and Kodaira dimension 2 surfaces are analogous to curves of genus . For this reason we introduce the following piece of terminology.
To complete the Enriques classification, we shall name all remaining possibilities for surfaces of Kodaira dimension 0 and 1.
However, the converse is not true: the ruled surface is elliptic but has Kodaira dimension . An abelian surface which is an extension of two elliptic curves is also elliptic but has Kodaira dimension 0 (because any abelian surface has trivial canonical bundle).
Hyperelliptic surfaces form another subclass of elliptic surfaces with . Now we are in a position to classify all surfaces with ([1, VIII.2]).
A huge amount of geometry of these surfaces have been discovered since the 19th century, which may be the topic of another (sketchy) note.
So far we have complete the Enriques classification of minimal algebraic surfaces:
And also some extra classes in the case of characteristic :
Complex algebraic surfaces, Cambridge University Press, 1996.
Complex Compact Surfaces, Springer, 2004.
Algebraic Geometry, Springer, 2010.