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

TL;DR

This paper investigates the properties of logarithmic bundles associated with multi-degree arrangements in complex projective space, establishing Torelli-type theorems and exploring moduli spaces, with specific results for arrangements involving lines and conics.

Contribution

It proves a Torelli-type theorem for large arrangements, describes moduli spaces for certain configurations, and relates bundles of different arrangements in $ extbf{P}^2$.

Findings

Reconstruction of arrangements as unstable hypersurfaces in the logarithmic bundle.

Certain arrangements of lines and conics are not Torelli.

Connections between bundles of different arrangements, including cubic and conic cases.

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.