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.

22 pages, 3 figures

References in corpus (1)

• The Torelli problem for Logarithmic bundles of hypersurface arrangements in the projective space