Free arrangements and coefficients of characteristic polynomials
  Ziegler showed that free arrangements have free restricted multiarrangements (multirestrictions). After Ziegler's work, several results concerning "reverse direction", namely characterizing freeness of an arrangement via that of multirestriction, have appeared. In this paper, we prove that the second Betti number of the arrangement plays a crucial role. 
Takuro Abe and Masahiko Yoshinaga
math.CO math.AC