Minimal number of singular fibers in a nonorientable Lefschetz fibration

TL;DR

This paper establishes the minimal conditions under which a nonorientable Lefschetz fibration can have only one singular fiber, linking the genus of the fiber and base surface.

Contribution

It proves the existence criteria for nonorientable genus g Lefschetz fibrations with a single singular fiber based on the genera of fiber and base surfaces.

Findings

Existence of such fibrations for g ≥ 4 and h ≥ 1

No such fibrations exist for lower g or h values

Provides a complete characterization of minimal singular fibers in this context

Abstract

We show that there exists an admissible nonorientable genus $g$ Lefschetz fibration with only one singular fiber over a closed orientable surface of genus $h$ if and only if $g \geq 4$ and $h \geq 1$.