Construction of bicuspidal rational complex projective plane curves

TL;DR

This paper presents explicit constructions of bicuspidal rational complex projective plane curves, detailing their singularity configurations and providing birational transformations from simpler curves, enhancing understanding of their geometric properties.

Contribution

It introduces explicit methods to construct bicuspidal rational curves with specified singularities and provides detailed singularity data and transformations from basic curves.

Findings

Explicit constructions of bicuspidal rational curves

Detailed Newton pairs and multiplicity sequences

Birational transformations from simple curves

Abstract

We give several constructions of bicuspidal rational complex projective plane curves, and list the Newton pairs and the multiplicity sequences of the singularities on the resulting curves. Although the existence of some of the listed cusp configurations is already proved in the literature, the advantage of our approach is that on one hand we explicitly state the Newton pairs and the multiplicity sequences of the singularities on the curves, on the other hand, we also give birational transformations transforming a conic or a bicuspidal curve with simple equation to these rational bicuspidal curves with given cusp configurations.