| This thesis takes some first steps towards understanding the geometry of the moduli spaces of stable surfaces. These moduli spaces are modular compactifications of the quasiprojective moduli spaces of minimal surfaces of general type considered by Tankeev, Artin, and Gieseker. They are, in light of higher-dimensional birational geometry, the higher-dimensional analogs of the moduli spaces of stable curves.; In contrast with stable curves, the moduli spaces of surfaces can be disconnected and quite singular (their associated moduli stacks need not be smooth). Some examples due to Catanese were known in this direction; this thesis provides some additional examples. Also, the geometry of the boundary of the moduli space is much more subtle than the case of moduli spaces of curves, where the boundary is a normal crossings divisor.; The thesis begins with several preliminary results on semi-log canonical singularities, which are the singularities allowed on stable surfaces. These results are used later to determine when certain surfaces are stable. Following this technical introduction, the moduli space of stable surfaces is defined. An example shows that in contrast to the case of curves, stablity of a family of surfaces does not follow from stability of its members; this example shows that if some additional conditions are not imposed on families, the moduli space would be nonseparated. Actually, there are two apparently different definitions for the moduli space of stable surfaces which are generally accepted. It remains unknown if they differ.; Following these technical preliminaries, the stable degenerations of products of smooth curves and surfaces isogenous to products of curves are determined. In the case of products of curves, the moduli space of such stable surfaces is constructed and completely determined. For surfaces isogenous to products, the results are applied to strengthen a result of Catanese about the moduli space of smooth surfaces isogenous to a product.; Continuing the study of surfaces built from curves, symmetric products are considered. In this case, the fibered symmetric square of a stable degeneration of curves is not a stable degeneration of surfaces. Locally, the minimal model program is applied to partially resolve the singularities of such a family. However, stability does not follow even for a family of surfaces obtained from a family of smooth curves, if the family has hyperelliptic members. The Hilbert scheme of length two subschemes of a curve is proposed as a stable reduction in the case of non-hyperelliptic curves. In general, it remains to be worked out how to "stablize" this Hilbert scheme in the case of nodal curves.; Finally, we prove that the closure of the locus in the moduli space parameterizing surfaces whose canonical model is singular is a divisor (if nonempty and proper). Some criteria are given for constructing a surface whose moduli point lies on the boundary and such that the boundary is high codimension at this point.; In a concluding chapter, several questions for future study are stated. |
