Rational cuspidal curves with four cusps on Hirzebruch surfaces

TL;DR

This paper investigates the possible number and types of cusps on rational cuspidal curves on Hirzebruch surfaces, constructing examples with four cusps using birational transformations and analyzing their properties.

Contribution

It introduces a method to construct rational cuspidal curves with four cusps on Hirzebruch surfaces and explores their characteristics.

Findings

Constructed explicit examples of rational cuspidal curves with four cusps.

Analyzed the properties and configurations of these curves.

Provided insights into the cusp configurations on Hirzebruch surfaces.

Abstract

The purpose of this article is to shed light on the question of how many and what kind of cusps a rational cuspidal curve on a Hirzebruch surface can have. We use birational transformations to construct rational cuspidal curves with four cusps on the Hirzebruch surfaces and find associated results for these curves.