LOGARITHMIC BUNDLES OF MULTI-DEGREE ARRANGEMENTS IN Pn

Abstract. Let D = {D1,..., D`} be a multi-degree arrangement with normal crossings on the complex projective space Pn, with degrees d1,..., d`; let Ω 1 Pn(logD) be the logarithmic bundle attached to it. First we prove a Torelli type theorem when D has a sufficiently large num-ber of components by recovering them as unstable smooth irreducible degree-di hypersurfaces of Ω 1 Pn(logD). Then, when n = 2, by describing the moduli spaces containing Ω1P2(logD), 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