Surface of general type

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

there is a quasi-projective scheme classifying the surfaces of general type with those Chern numbers.

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.

[1] 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.

the minimum possible value for general type, and surfaces on the line

These surface which are located in the "lower left" boundary in the diagram have been studied in detail.

Surfaces with all these values are known; a few of the many examples that have been studied are: Bombieri (1973) proved that the multicanonical map φnK for a complex surface of general type is a birational isomorphism onto its image whenever n≥5, and Ekedahl (1988) showed that the same result still holds in positive characteristic.