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

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 $Ω_{\mathbf{P}^{n}}^{1}(\log \mathcal{D})$ 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 $Ω_{\mathbf{P}^{n}}^{1}(\log \mathcal{D})$. Then, when $n = 2$, by describing the moduli spaces containing $Ω_{\mathbf{P}^{2}}^{1}(\log \mathcal{D})$, 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.

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