Dihedral Monodromy and Xiao Fibrations

TL;DR

This paper constructs new algebraic surface fibrations with dihedral monodromy that challenge Xiao's conjecture, and demonstrates the existence of special divisors in the Brill-Noether range.

Contribution

It introduces three new families of fibrations with dihedral monodromy that violate Xiao's conjecture and shows the existence of big and nef divisors in the Brill-Noether range.

Findings

Fibrations violate Xiao's conjecture on irregularity and fiber genus.

Existence of big and nef divisors in the Brill-Noether range.

Construction of covers with dihedral monodromy.

Abstract

We construct three new families of fibrations $\pi : S \to B$ 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 $\pi$ are certain \'etale 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.