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 of π are certain étale cyclic covers of hyperelliptic curves that give coverings of P 1 with dihedral monodromy. As an application, we also show the existence of big and nef effective divisors in the Brill–Noether range.