A note on the rational cuspidal curves

TL;DR

This paper provides an elementary combinatorial proof that a conjecture related to rational cuspidal curves follows from existing results in the case of curves with two critical points.

Contribution

It establishes a link between a conjecture and prior results specifically for rational cuspidal curves with two critical points using a combinatorial approach.

Findings

The conjecture holds for rational cuspidal curves with two critical points.

An elementary combinatorial argument suffices to derive the conjecture from known results.

The paper clarifies the relationship between the conjecture and existing theorems in this specific case.

Abstract

In this short note we give an elementary combinatorial argument, showing that the Conjecture of J. Fern\'andez de Bobadilla, I. Luengo, A. Melle-Hern\'andez, A. N\'emethi follows from the results of M. Borodzik and C. Livingston in the case of rational cuspidal curves with two critical points.