Freeness of conic-line arrangements in $P^2$

Abstract

Let $\mathcal{C}={\bigcup }_{i=1}^n{C}_i\subseteq P^2$ 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 ${\Omega }^1(\mathcal{C})$ of logarithmic differential forms with pole along $\mathcal{C}$. We also show that the analog of Terao’s conjecture (freeness of ${\Omega }^1(\mathcal{C})$ is combinatorially determined if $\mathcal{C}$ is a union of lines) is false in this setting.