Deletion-addition of a smooth conic for free curves

TL;DR

This paper investigates how removing or adding a smooth conic affects free reduced plane projective curves, providing new insights into their geometric and combinatorial properties, especially for conic-line arrangements.

Contribution

It introduces new results on the behavior of free curves under conic deletion and addition, extending known results from line arrangements to more general curves.

Findings

Obstructions to the geometry of free reduced curves

Generalization of results from line arrangements to conic-line arrangements

Insights into the combinatorial structure of free curves

Abstract

We describe the behaviour of a free reduced plane projective curve with respect to the deletion, respectively addition, of a smooth conic. These results apply in particular to conic-line arrangements. We present some obstructions to the geometry and combinatorics of a free reduced curve, generalizing results known a priori only for free projective line arrangements.