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

Abstract

Let C = ∪ni = 1 Ci ⊆ ℙ2 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 Ω1(C) of logarithmic differential forms with pole along C. We also show that the analog of Terao’s conjecture (freeness of Ω1(C) is combinatorially determined if C is a union of lines) is false in this setting.