New results and problems on moduli spaces of algebraic surfaces

Fabrizio Catanese (Bayreuth)

In my talk I will focus on recent results, obtained in collaboration with Ingrid Bauer, concerning the explicit determination of moduli spaces of certain surfaces with genus zero, in particular of the so called Burniat surfaces.

These results shed light on interesting open problems relating the several moduli spaces for these geometric objects: the Gieseker moduli spaces for canonical models, the moduli spaces for minimal models, and the compactified moduli spaces for stable surfaces. Here, I will mention recent results by Rollenske and Wenfei Liu concerning the connected components of these compactifications.

Interesting questions, motivated by the results of Vakil concerning `Murphy's law'

for the singularities of moduli spaces , and the example of nodal Burniat surfaces of canonical degree 4 and 3, concern the relation between the local structures of the above first two mentioned moduli spaces, and a similar relation for the deformations of automorphisms.