Dihedral monodromy and Xiao fibrations

We construct three new families of fibrations π:S→B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi : S \rightarrow B$$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi $$\end{document} are certain étale cyclic covers of hyperelliptic curves that give coverings of P1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {P}}^1$$\end{document} with dihedral monodromy. As an application, we also show the existence of big and nef effective divisors in the Brill–Noether range.