Final Report Summary - SURFARI (Arithmetic of algebraic surfaces) For an algebraic surface, the Picard number is arguably the most basic invariant which is not preserved under deformations. Roughly, it measures the quantity of curves lying on the surface, thus providing a deep insight into its inner structures. The project is concerned with the fundamental problem which Picard numbers occur within a given class of surfaces.Or main result is an affirmative solution for complex quintics in IP^3: all Picard numbers from 1 to 45 do actually occur. Key ingredients consist in a novel technique of arithmetic deformations, reductions to K3 surfaces and wild automorphisms. Towards higher degrees, we developed a closed formula for the Picard numbers of Delsarte surfaces with isolated rational double point singularities. For sextics in IP^3, this enables us to realize most small Picard numbers in a rather explicit way.Another project deals with the more refined problem of singular complex K3 surfaces, i.e. those attaining the maximum Picard number 20. Here the difference between field of definition and field of moduli is the critical point. We analyzed the fields of moduli completely and proved that all singular K3 surfaces of class number 2 possess a model defined over Q.The third main stream of research concerns Enriques surfaces. We give a complete classification of Enriques surfaces supporting a maximal root type generated by smooth rational curves. Remarkably, the techniques can be developed quite uniformly for all characteristics, and the underlying moduli spaces behave very nicely.All in all, the project has facilitated many interesting results and powerful novel techniques for the study of the arithmetic of algebraic surfaces and their Picard groups.
