Logarithmic bundles of multi-degree arrangements in P^n

Let D = {D_1, . . . ,D_ℓ} be a multi-degree arrangement with normal crossings on the complex projective space P^n, with degrees d_1, . . . , d_ℓ; let consider the logarithmic bundle attached to it. First we prove a Torelli type theorem when D has a sufficiently large number of components by recovering them as unstable smooth irreducible degree-d_i hypersurfaces of the logarithmic bundle. Then, when n = 2, by describing the moduli spaces containing the logarithmic bundle, we show that arrangements of a line and a conic, or of two lines and a conic, are not Torelli. Moreover we prove that the logarithmic bundle of three lines and a conic is related with the one of a cubic. Finally we analyze the conic-case.