Zariski Pairs Beyond Topology: Arithmetic and Real Aspects

Multicanonically embedded surfaces in projective space give rise, via projection from generic centers, to irreducible branch curves in the plane. Building on our previous work, we transfer results from the moduli space of surfaces of general type to equisingular strata of plane curves. In particular, the faithful action of the Galois group on the connected components of moduli spaces of surfaces isogenous to a product, as established by Bauer, Catanese, and Grunewald, produces large families of arithmetic Zariski multiplets. We further discuss how these constructions can be adapted to the real setting, leading to examples of real Zariski pairs and multiplets, and highlighting the interplay between arithmetic, topology, and real structures. This is joint work in progress with M. Lönne.