Deletion-addition of a smooth conic for free curves

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.

💡 Research Summary

The paper investigates how the freeness property of a reduced plane curve C behaves under the deletion or addition of a smooth conic Q. The authors focus especially on conic‑line arrangements (CL‑arrangements), which are natural extensions of line arrangements, and they aim to generalize the well‑known addition‑deletion theorems from hyperplane arrangement theory to the setting of plane curves.

First, the authors recall the definition of a free curve: the module of logarithmic derivations D(C) is a free S‑module, where S = ℂ