Logarithmic bundles of hypersurface arrangements in Pn

Let be an arrangement of smooth hypersurfaces with normal crossings on the complex projective space and let be the logarithmic bundle attached to it. Following (Ancona in Notes of a talk given in Florence, 1998), we show that admits a resolution of length which explicitly depends on the degrees and on the equations of . Then we prove a Torelli type theorem when all the 's have the same degree and : indeed, we recover the components of as unstable smooth hypersurfaces of . Finally we analyze the cases of one quadric and a pair of quadrics, which yield examples of non-Torelli arrangements. In particular, through a duality argument, we prove that two pairs of quadrics have isomorphic logarithmic bundles if and only if they have the same tangent hyperplanes.