Surfaces of general type

In algebraic geometry, a surface of general type is an algebraic surface with Kodaira dimension 2. Because of Chow's theorem any compact complex manifold of dimension 2 and with Kodaira dimension 2 will actually be an algebraic surface, and in some sense most surfaces are in this class.

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. It remains a very difficult problem to describe these schemes explicitly, and there are 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 complex surface of general type must satisfy:

Many (and possibly all) pairs of integers satisfying these conditions are the Chern numbers for some complex surface of general type. By contrast, for almost complex surfaces, the only constraint is:

and this can always be realized.

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. In particular 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, and surfaces on the line 3c₂ = c₁² are all quotients of the unit ball in C² (and are particularly hard to find).

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