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

Abstract

Let $D=D_1,\dots ,D_{\ell }$ be a multi-degree arrangement with normal crossings on the complex projective space $P^n$, with degrees $d_1,\dots ,d_{\ell }$; let ${\Omega }_{P^n}^1(\log D)$ be the logarithmic bundle attached to it. First we prove a Torelli type theorem when $D$ has a sufficiently large number of components by recovering them as unstable smooth irreducible degree-$d_i$ hypersurfaces of ${\Omega }_{P^n}^1(\log D)$. Then, when $n=2$, by describing the moduli spaces containing ${\Omega }_{P^2}^1(\log D)$, 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.