Topological obstructions for rational cuspidal curves in Hirzebruch surfaces

TL;DR

This paper investigates the existence of rational cuspidal curves in Hirzebruch surfaces, introducing two new obstructions based on Heegaard--Floer theory and spectral analysis of links at infinity.

Contribution

It provides novel obstructions for rational cuspidal curves in Hirzebruch surfaces, extending previous results with new theoretical tools.

Findings

Heegaard--Floer theory yields a generalization of Livingston and the first author's result.

Spectral comparison offers a new criterion for curve existence.

The paper establishes conditions that restrict possible singular point configurations.

Abstract

We study rational cuspidal curves in Hirzebruch surfaces. We provide two obstructions for the existence of rational cuspidal curves in Hirzebruch surfaces with prescribed types of singular points. The first result comes from Heegaard--Floer theory and is a generalization of a result by Livingston and the first author. The second criterion is obtained by comparing the spectrum of a suitably defined link at infinity of a curve with spectra of its singular points.