Dihedral Monodromy and Xiao Fibrations

We construct three new families of fibrations $π: S \to B$ 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.

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