On the nearly freeness of conic-line arrangements with nodes, tacnodes, and ordinary triple points

TL;DR

This paper classifies nearly free conic-line arrangements with specific singularities in the complex plane, establishing degree bounds and constructing examples for degrees 3 to 7, while proving non-existence for degrees 10 to 12.

Contribution

It provides a partial classification and degree bounds for nearly free conic-line arrangements with nodes, tacnodes, and triple points, including explicit examples and non-existence results.

Findings

Degree of such arrangements is bounded above by 12.

Examples exist for degrees 3 to 7.

No arrangements exist for degrees 10 to 12.

Abstract

In the present note we provide a partial classification of nearly free conic-line arrangements in the complex plane having nodes, tacnodes, and ordinary triple points. In this setting, our theoretical bound tells us that the degree of such an arrangement is bounded from above by $12$. We construct examples of nearly free conic-line arrangements having degree $3,4,5,6,7$, and we prove that in degree $10$, $11$, and $12$ there is no such arrangement.