Construction of free curves by adding osculating conics to a given cubic curve

TL;DR

This work addresses the problem of constructing free and nearly free plane curve arrangements by augmenting a plane cubic with hyperosculating conics. It develops a systematic analysis for two classes of cubics: a nodal cubic and the Fermat cubic, combining Jacobian syzygies, Tjurina numbers, and Hessian-based detectors of hyperosculating conics. For nodal cubics, it completely classifies arrangements with up to three hyperosculating conics, yielding free exponents $(2,2)$, $(3,3)$, and a $4$-syzygy case with $(5,5,5,5)$ (the latter being a maximal Tjurina curve with $(d,r)=(9,5)$); a nearly free example with exponents $(3,4)$ is also exhibited. For the Fermat cubic, the 27 hyperosculating conics are organized into 9 symmetry classes under group actions, enabling precise freeness outcomes: two conics in the same class give $(3,3)$, three conics in the same class give $(3,5)$, while mixing classes can yield nearly free $(3,4)$ configurations; a detailed local analysis confirms these results, including a $J_{2,0}$-type singularity for the EC$_3$ case. The results expand the catalog of free and nearly free plane curves via high-order tangency configurations and symmetry, with substantial computational verification in ${ m Singular}$.

Abstract

In the present article we construct new families of free and nearly free curves starting from a plane cubic curve $C$ and adding some of its hyperosculating conics. We present results that involve nodal cubic curves and the Fermat cubic. In addition, we provide new insight into the geometry of the $27$ hyperosculating conics of the Fermat cubic curve using well-chosen group actions.

Theorems & Definitions (28)

• Definition 2.1
• Remark 2.2
• Definition 2.3
• Theorem 2.4: Dmax
• Theorem 2.5: Dmax
• Remark 2.6
• Remark 2.7
• Definition 2.8
• Definition 2.9: Cayley
• Theorem 3.1: Coolidge