On the freeness of rational cuspidal plane curves

TL;DR

This paper provides evidence supporting the conjecture that rational cuspidal plane curves are either free or nearly free, confirming it for degrees up to 34 and many odd degrees.

Contribution

It extends the verification of the conjecture to many odd degrees and degrees up to 34, advancing understanding of the structure of rational cuspidal plane curves.

Findings

Conjecture holds for degrees up to 34.

Confirmed for many odd degrees.

Supports the conjecture that such curves are free or nearly free.

Abstract

We bring additional support to the conjecture saying that a rational cuspidal plane curve is either free or nearly free. This conjecture was confirmed for curves of even degree, and in this note we prove it for many odd degrees. In particular, we show that this conjecture holds for the curves of degree at most 34.