Logarithmic bundles of hypersurface arrangements in P^n

Let $ \mathcal{D}=\{D_{1}, \ldots, D_{\ell}\} $ be an arrangement of smooth hypersurfaces with normal crossings on the complex projective space $ \mathbf{P}^n $ and let $ \Omega^{1}_{\mathbf{P}^n}(\log \mathcal{D}) $ be the logarithmic bundle attached to it. Following \cite{AppuntiAncona}, we show that $ \Omega^{1}_{\mathbf{P}^n}(\log \mathcal{D}) $ admits a resolution of lenght $ 1 $ which explicitly depends on the degrees and on the equations of $ D_{1}, \ldots, D_{\ell} $. Then we prove a Torelli type theorem when all the $ D_{i} $'s have the same degree $ d $ and $ \ell \geq {{n+d} \choose {d}}+3 $: indeed, we recover the components of $ \mathcal{D} $ as unstable smooth hypersurfaces of $ \Omega^{1}_{\mathbf{P}^n}(\log \mathcal{D}) $. 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.