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

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.