A new hierarchy for complex plane curves

We define the type of a plane curve as the initial degree of the corresponding Bourbaki ideal. Then we show that this invariant behaves well with respect to the union of curves. Curves of type $0$ are precisely the free curves, while curves of type $1$ are the plus-one generated curves. In this paper, we first show that line arrangements and conic-line arrangements can exhibit all the theoretically possible types. In the second part, we study the properties of the curves of type $2$ and construct families of line arrangements and conic-line arrangements of this type.