Freeness of Conic-Line Arrangements in $\mathbb P^2$

Abstract

Let ${\mathcal C}= \bigcup_{i=1}^n C_i \subseteq \mathbb{P}^2$ be a collection of smooth rational plane curves. We prove that the addition-deletion operation used in the study of hyperplane arrangements has an extension which works for a large class of arrangements of smooth rational curves, giving an inductive tool for understanding the freeness of the module $Ω^1({\mathcal C})$ of logarithmic differential forms with pole along ${\mathcal C}$. We also show that the analog of Terao's conjecture (freeness of $Ω^1({\mathcal C})$ is combinatorially determined if ${\mathcal C}$ is a union of lines) is false in this setting.