Logarithmic bundles of hypersurface arrangements in $$\mathbf{P}^n$$ P n

Abstract

Let $${\mathcal {D}}=\{D_{1}, \ldots , D_{\ell }\}$$ be an arrangement of smooth hypersurfaces with normal crossings on the complex projective space $$\mathbf{P}^n$$ and let $$\Omega ^{1}_{\mathbf{P}^n}(\log {\mathcal {D}})$$ be the logarithmic bundle attached to it. Following (Ancona in Notes of a talk given in Florence, 1998), we show that $$\Omega ^{1}_{\mathbf{P}^n}(\log {\mathcal {D}})$$ admits a resolution of length $$1$$ which explicitly depends on the degrees and on the equations of $$D_{1},\ldots ,D_{\ell }$$ . Then we prove a Torelli type theorem when all the $$D_{i}$$ ’s have the same degree $$d$$ and $$\ell \ge {{n+d}\atopwithdelims (){d}}+3$$ : indeed, we recover the components of $${\mathcal {D}}$$ as unstable smooth hypersurfaces of $$\Omega ^{1}_{\mathbf{P}^n}(\log {\mathcal {D}})$$ . Finally we analyze the cases of one quadric and a pair of quadrics, which yield examples of non-Torelli arrangements. In particular, through a duality argument, we prove that two pairs of quadrics have isomorphic logarithmic bundles if and only if they have the same tangent hyperplanes.