Logarithmic bundles of multi-degree arrangements in $\mathbf{P}^{n}$

Abstract

Let $ \mathcal{D} = \{D_{1}, ..., D_{\ell}\} $ be a multi-degree arrangement with normal crossings on the complex projective space $ \mathbf{P}^{n} $, with degrees $ d_{1}, ..., d_{\ell} $; let $ \Omega_{\mathbf{P}^{n}}^{1}(\log \mathcal{D}) $ be the logarithmic bundle attached to it. First we prove a Torelli type theorem when $ \mathcal{D} $ has a sufficiently large number of components by recovering them as unstable smooth irreducible degree-$ d_{i} $ hypersurfaces of $ \Omega_{\mathbf{P}^{n}}^{1}(\log \mathcal{D}) $. Then, when $ n = 2 $, by describing the moduli spaces containing $ \Omega_{\mathbf{P}^{2}}^{1}(\log \mathcal{D}) $, 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.