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# Surface of general type

In algebraic geometry, a surface of general type is an algebraic surface with Kodaira dimension 2. These are all algebraic, and in some sense most surfaces are in this class.

## Classification

Gieseker showed that there is a coarse moduli scheme for surfaces of general type; this means that for any fixed values of the Chern numbers c12 and c2, there is a quasi-projective scheme classifying the surfaces of general type with those Chern numbers. However is a very difficult problem to describe these schemes explicitly, and there are very few pairs of Chern numbers for which this has been done (except when the scheme is empty!) There are some indications that these schemes are in general too complicated to write down explicitly: the known upper bounds for the number of components are very large, some components can be non-reduced everywhere, components may have many different dimensions, and the few pieces that have been studied explicitly tend to look rather complicated.

The study of which pairs of Chern numbers can occur for a surface of general type is known as "geography of Chern numbers" and there is an almost complete answer to this question. There are several conditions that the Chern numbers of a minimal surface of general type must satisfy:

• $c_1^2+c_2 equiv 0 pmod\left\{12\right\}$ (as it is equal to 12χ)
• c12 > 0, c2 > 0
• c12 ≤ 3c2 (proved by Miyaoka and Yau)
• 5c12c2 + 36 ≥ 0 (The Noether inequality)

Most (and possibly all) pairs of integers satisfying these conditions are the Chern numbers for some surface of general type; in fact it is known that the only possible exceptions lie in a strip near the line c12 = 3c2.

By contrast, for almost complex surfaces, the only constraint is $c_1^2+c_2 equiv 0 pmod\left\{12\right\}$, and this can always be realized.

## Examples

This is only a small selection of the rather large number of examples of surfaces of general type that have been found.

Many of the surfaces of general type that have been investigated lie on (or near) the edges of the region of possible Chern numbers:

• Horikawa surfaces lie on or near the "Noether line".
• Many of the surfaces listed below lie on the line c2+c12=12χ=12, the minimum possible value for general type.
• Surfaces on the line 3c2=c12 are all quotients of the unit ball in C2 (proved by Yau). They are particularly hard to find.

### Barlow surface

These are simply connected, and are the only known examples of simply connected surfaces of general type with pg=0. They are named for Rebecca Barlow.

Hodge diamond:

1
00
090
00
1

### Beauville surfaces

Fundamental group is infinite. They are named for Arnaud Beauville.

Hodge diamond:

1
00
020
00
1

### Campedelli surfaces

Hodge diamond:
1
00
080
00
1
Surfaces with the same Hodge numbers are called numerical Campedelli surfaces.

### Castelnuovo surfaces

Surfaces of general type such that the canonical bundle is very ample and such that c12=3pg-7. (Castelnuovo proved that if the canonical bundle is very ample for a surface of general type then c12≥3pg-7.)

### Catanese surfaces

Simply connected.

Hodge diamond:

1
00
080
00
1

### Complete intersections

A smooth complete intersection of hypersurfaces of degrees d1d2≥...≥dn-2 in Pn is a surface of general type unless the degrees are (2), (3), (2,2) (rational), (4), (3,2), (2,2,2) (Kodaira dimension 0). For example, in three-dimensional projective space, non-singular surfaces of degree at least 5 are of general type.

Complete intersections are all simply connected.

### Fake projective plane (Mumford surface)

This was found by Mumford using p-adic geometry. It has the same betti numbers as the projective plane, but is not homeomorphic to it as its fundamental group is infinite.

Hodge diamond:

1
00
010
00
1

### Godeaux surfaces

The cyclic group of order 5 acts freely on the Fermat surface of points (w:x:y:z) in P3 satisfying w5+x5+y5+z5=0 by mapping (w:x:y:z) to (w:ρx:ρ2y:ρ3z) where ρ is a fifth root of 1. The quotient by this action is the original Godeaux surface. The eponym was the Belgian geometer Lucien Godeaux.

Other surfaces constructed in a similar way with the same Hodge numbers are also sometimes called Godeaux surfaces. Surfaces with the same Hodge numbers (such as Barlow surfaces) are called numerical Godeaux surfaces.

The fundamental group (of the original Godeaux surface) is cyclic of order 5.

Hodge diamond:

1
00
090
00
1

### Hilbert modular surfaces

These are mostly of general type.

### Horikawa surfaces

These are surfaces with q=0 and pg=c12/2 +2 or c12/2+3/2 (which implies that they are more or less on the "Noether line" edge of the region of possible values of the Chern numbers). They are all simply connected, and Horikawa gave a detailed description of them.

### Products

The product of two curves both of genus at least 2 is a surface of general type.