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