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

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Let \mathcal C = \bigcup^n_{i = 1} C_i ⊆ 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.