Seminario di Geometria Algebrica: Logarithmic bundles of multi-degree arrangements in P^n (Elena Angelini)

Speaker: Elena Angelini (Università di Ferrara)

Title: Logarithmic bundles of multi-degree arrangements in $P^n$

Abstract: Let D = {D_{1},...,D_{l}} be a multi-degree arrangement of smooth hypersurfaces with normal crossings on the complex projective space P^n and let O^{1}_P^n(logD) be the logarithmic bundle attached to it. We show that O^{1}_P^n(logD) admits a resolution of length 1 which explicitly depends on the degrees and on the equations of D_{1},...,D_{l}. Then we prove a Torelli type theorem when D is made of a "sufficiently" large number of hypersurfaces: indeed, we recover the components of D as unstable smooth hypersurfaces of D_{1},...,D_{l}. Moreover we analyze the case 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. Finally we give a description of the conic case and of some line-conic cases.