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 [1]. See also [2] and [3].

## Blow-ups and birational maps

Let be a surface. The structure of rational maps between and any projective variety is simple by the following

Theorem 1 (Elimination of indeterminacy) Let be a rational map. Then there exists a composite of finite many blow-ups and a morphism such that .
Proof We have a bijection between nondegenerate rational maps and linear systems on of dimension which has no fixed components. We use this correspondence and induct on the dimension of the linear systems. Each blow-up drops the dimension and this process terminates using the intersection pairing. ¡õ

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

Theorem 2 Let be a birational map between surfaces. Then there exists morphisms and such that , where each of can be decomposed as a sequence of blow-ups and an isomorphism.
Example 1 (Quadric surfaces) Let be a smooth quadric. The projection from a point gives a birational map . Let be the blow-up of at . We can identify as The composition coincides with the projection onto the second factor , which is also the blow-up of at the two points corresponding to the two lines on passing through .

## Neron-Severi groups and minimal models

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 .

Definition 1 The group is called the Neron-Severi group 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.

Theorem 3 Let be a blow-up at a point with exceptional divisor . Then there is a canonical isomorphism .
Proof The canonical isomorphism given by descends to the Neron-Severi groups. ¡õ

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 .

Definition 2 Let . We say that dominates if there exists a birationally morphism (in particular, ). We say that is minimal if every birational morphism is an isomorphism.

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]).

Theorem 4 (Castelnuovo's contractibility criterion) Suppose a curve is isomorphic to with . Then is an exceptional curve on .

## Ruled surfaces

In most cases, has a unique minimal element. However, the situation is not that simple for a large class of surfaces — ruled surfaces.

Definition 3 A surface is called ruled if it is birational to for a smooth projective curve.
Example 2 is a ruled surface. More generally, for any vector bundle of rank two over the curve , the projective bundle associated to is a ruled surface due to the local trivialization. Every rational surface is ruled as it is birational to , hence birational to .

As a little digression, we can calculate several birational invariants for ruled surfaces easily.

Definition 4 Let be any surface. We define
• the irregularity . It is equal to by Hodge theory. It is the dimension of the Picard variety and the Albanese variety of . It is also the difference of the geometric genus and the arithmetic genus, hence its name.
• the geometric genus . It is equal to by Serre duality and by Hodge theory.
• the plurigenus ().
Theorem 5 Let be a ruled surface over . Then , and for all .
Proof Since these are birational invariants, we may assume . Using the isomorphism , we know that . Using the Künneth formula , we know that for all . ¡õ

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.

Definition 5 A geometrically ruled surface over is a surface together with a smooth morphism whose fibers are isomorphic to .

The first thing to notice is that geometrically ruled surfaces form a subclass of ruled surfaces due to the following theorem ([1, III.4]).

Theorem 6 (Noether-Enriques) Let be a surface together with a smooth morphism . If for some point , is smooth over and is isomorphic to . Then there exists an open neighborhood of such that is a trivial bundle. In particular, every geometrically ruled surface is ruled.

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.

Theorem 7 Let be a smooth irrational curve. Then the minimal models of are geometrically rules surfaces over .
Proof Suppose is a minimal surface and is a birational map. Then by elimination of indeterminacy, we can find another surface fitting in the following diagram where is the projection onto the first factor and is a composition of blow-ups. Suppose is the smallest such integer. If , let be the exceptional curve of the -th blow-up, then must be a point since is irrational by assumption. So we can eliminate the -th blow-up, which contradicts the minimality of . Therefore and is actually a morphism with its generic fiber isomorphic to . Hence ([1, III.8]) is a geometrically ruled surface over . ¡õ

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.

## Rational surfaces and Castelnuovo's theorem

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]).

Proposition 1 If , then there is a unique irreducible curve on with negative self-intersection. Moreover, its self-intersection is .

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 ().

Lemma 1 Let be a minimal surface with . Then there exists a smooth rational curve on such that .

Now we can deduce the following classification of minimal models of rational surfaces.

Theorem 8 The minimal rational surfaces are the Hirzebruch surfaces () and itself.
Proof Let be a minimal rational surfaces. By the lemma, there exists smooth curves on with the least nonnegative . Choose such a curve with the least , where is a hyperplane section of . Then using the minimality and Riemann-Roch, one can show that every divisor is a smooth rational curve. Since the linear system of curves of passing through with multiplicity has codimension in . We know that . Suppose , then for any , the exact sequence implies that has no base point and as . Therefore . When , the morphism is geometrically ruled over , hence is for some . When , each fiber of the morphism is the intersection of two distinct rational curves, hence a point. Therefore . ¡õ

A similar argument using above useful lemma implies the following numerical characterization of rational surfaces ([1, V.1]).

Theorem 9 (Castelnuovo's Rationality Criterion) Let be a surface with . Then is rational.

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]).

Theorem 10 Let be two minimal non-ruled surfaces. Then every birational map between and is an isomorphism. In particular, every non-ruled surface admits a unique minimal model.

Hence we have found a complete list of minimal surfaces by now.

## Kodaira dimension

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]).

Theorem 11 (Enriques) Let be a surfaces with (or ). Then is ruled. In particular, for all .

In view of the important role played by the plurigenera, we introduce the notion of Kodaira dimension for a smooth projective variety.

Definition 6 Let be a smooth projective variety and be the rational map from to the projective space associated to the complete linear system . We define the Kodaira dimension of to be the maximal dimension of the images of for . We write if for .

In particular, the Kodaira dimension is always no greater than the dimension of . Some examples are in order.

Example 3 Let be a curve of genus . Then Riemann-Roch implies the following correspondence:
• : ,
• : ,
• : .
Example 4 Let be a surface. Then we have the following correspondence:
• : for . By Enriques' theorem, these are exactly the ruled surfaces. Moreover, those with are exactly the rational surfaces,
• : (not all 0),
• : and is a curve for some ,
• : and is a surface for some .
Example 5 Let be two curves and . From the example of curves, we know that
• : or ,
• : ,
• : or ,
• : .

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.

Definition 7 We say that a surface is of general type if .
Example 6 Let be a surface in which is a complete intersection of hypersurfaces of degrees . Then as an application of the adjunction formula, we know that
• : , , ,
• : , , ,
• : all other cases.

## Enriques classification

To complete the Enriques classification, we shall name all remaining possibilities for surfaces of Kodaira dimension 0 and 1.

Definition 8 A surface is called elliptic if is equipped with an elliptic fibration over some smooth curve , i.e., there is a surjective morphism whose generic fiber is an elliptic curve.
Theorem 12 Let be a minimal surface with . Then is elliptic ([1, IX.2]).

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).

Definition 9 A surface is called hyperelliptic (or bielliptic) if , where , are two elliptic curves and is a finite group of translations of acting on such that .

Hyperelliptic surfaces form another subclass of elliptic surfaces with . Now we are in a position to classify all surfaces with ([1, VIII.2]).

Theorem 13 Let be a minimal surfaces with . Then there are four possibilities:
• , ; these are called Enriques surfaces,
• , ; these are hyperelliptic surfaces,
• , ; these are called K3 (Kummer-Kähler-Kodaira) surfaces,
• , ; these are abelian surfaces.
Remark 1 It can be shown that all Enriques surfaces are elliptic. There are also some examples of elliptic K3 surfaces.

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:

• : ruled surfaces (including rational surfaces),
• : Enriques surfaces, hyperelliptic surfaces, K3 surfaces, abelian surfaces,
• : elliptic surfaces (excluding the above two cases),
• : surfaces of generic type.
Remark 2 We end this note by remarking that there are also non-algebraic compact complex surfaces which have been classified by Kodaira:
• : surfaces of class VII,
• : complex tori (non-algebraic abelian surfaces), non-algebraic K3 surfaces, primary and secondary Kodaira surfaces,
• : non-algebraic elliptic surfaces.

And also some extra classes in the case of characteristic :

• : non-classical Enriques surfaces (), quasi-hyperelliptic (),
• : quasi-elliptic surfaces ()

#### References

[1]Beauville, A., Complex algebraic surfaces, Cambridge University Press, 1996.

[2]W. Barth and K. Hulek and Chris Peters and A.van de Ven, Complex Compact Surfaces, Springer, 2004.

[3]Robin Hartshorne, Algebraic Geometry, Springer, 2010.