Dihedral monodromy and Xiao fibrations

We construct three new families of fibrations π : S → B where S is an algebraic
complex surface and B a curve that violate Xiao’s conjecture relating the relative irregularity
and the genus of the general fiber.The fibers ofπ are certain étale cyclic covers of hyperelliptic
curves that give coverings of P1 with dihedral monodromy. As an application, we also show
the existence of big and nef effective divisors in the Brill–Noether range.