The Scrollar Invariants of Curves Mapping to a Hirzebruch Surface

TL;DR

This paper studies the scrollar invariants of certain covers of the projective line that factor through nodal curves in Hirzebruch surfaces, and establishes an existence theorem for such curves with specified properties.

Contribution

It introduces a detailed analysis of scrollar invariants for covers factoring through Hirzebruch surface nodal curves and proves an existence theorem for these curves with fixed class and singularities.

Findings

Characterization of scrollar invariants for covers factoring through Hirzebruch surface curves

Existence theorem for nodal curves with prescribed class and singular locus in Hirzebruch surfaces

Insights into the structure of covers and their invariants in algebraic geometry

Abstract

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.