On a class of rational cuspidal plane curves

TL;DR

This paper classifies certain rational cuspidal plane curves with multiple cusps, providing new examples and demonstrating their projective rigidity, while discussing the broader context of their rigidity properties.

Contribution

It offers a complete classification of rational cuspidal plane curves with at least three cusps and one with high multiplicity, highlighting their projective rigidity.

Findings

Complete list of such curves with three or more cusps

Identification of projective rigidity in these curves

Discussion on the general rigidity problem

Abstract

We obtain new examples and the complete list of the rational cuspidal plane curves $C$ with at least three cusps, one of which has multiplicity ${\rm deg}\,C - 2$. It occurs that these curves are projectively rigid. We also discuss the general problem of projective rigidity of rational cuspidal plane curves.