On rational cuspidal plane curves, and the local cohomology of Jacobian rings

TL;DR

This paper classifies rational cuspidal plane curves of degree at least 6 with weighted homogeneous singularities, explores their relation to free curves via Tjurina numbers, and proposes a stronger version of Terao's conjecture.

Contribution

It provides a complete projective classification of certain rational cuspidal plane curves and offers new insights into their freeness properties and Tjurina numbers.

Findings

Complete classification of rational cuspidal plane curves of degree ≥6

New characterization of free and nearly free curves via Tjurina numbers

Proposal of a stronger form of Terao's conjecture

Abstract

This note gives the complete projective classification of rational, cuspidal plane curves of degree at least 6, and having only weighted homogeneous singularities. It also sheds new light on some previous characterizations of free and nearly free curves in terms of Tjurina numbers. Finally, we suggest a stronger form of Terao's conjecture on the freeness of a line arrangement being determined by its combinatorics.