The Scrollar Invariants of Curves Mapping to a Hirzebruch Surface

In this note we analyse the scrollar invariants of $k:1$ covers of ${\mathbb{P}}^1$ that factor through the normalisation of a nodal curve in the $m$-th Hirzebruch surface ${\mathbb{F}}_m$. We then give an existence theorem for nodal curves in ${\mathbb{F}}_m$ having fixed class and singular locus.