Surfaces of general type

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${\displaystyle c_{1}^{2},c_{2},}$ 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.