A new hierarchy for complex plane curves

TL;DR

This paper introduces a new hierarchy for complex plane curves based on the initial degree of their Bourbaki ideal, classifies free and plus-one generated curves within this hierarchy, and explores properties of curves of type 2.

Contribution

It defines a novel invariant for plane curves, analyzes its behavior under union, and constructs examples of arrangements for different types, especially type 2.

Findings

Curves of type 0 are exactly the free curves.

Curves of type 1 are the plus-one generated curves.

All theoretically possible types are realized by line and conic-line arrangements.

Abstract

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.