Hyperplane arrangements of Torelli type

Abstract

We give a necessary and sufficient condition in order for a hyperplane arrangement to be of Torelli type, namely that it is recovered as the set of unstable hyperplanes of its Dolgachev sheaf of logarithmic differentials. Decompositions and semistability of non-Torelli arrangements are investigated.Comment: 2 Figue

Logarithmic bundles and line arrangements, an approach via the standard construction

“…Proof. This is somehow implicit in [FMV10,FV12]), still we give here a full proof. Let us consider the canonical exact sequence of coherent sheaves of ˇ n -modules:…”

“…Proof. This is somehow implicit in [3,4], but we give here a simplified proof. First of all, let H be a hyperplane in P n .…”

“…Assume that A is free with exponents (k, k + r), with r 0, k 1, that A has points of multiplicity 3 at most, and that not all lines of A pass through a point. Then the possible pairs (k, k + r) are (1, 1), (1, 2), (2, 2), (2, 3), (3,3), (3,4) or (4,4). In the last case, A has the combinatorial type of the dual Hesse arrangement.…”

A Torelli-Type Problem for Logarithmic Bundles Over Projective Varieties

“…Look at the conormal exact sequence of ν d (X) in P r : (7) and then restricting it to D := ν d (E) we get the exact sequence…”

On stability of logarithmic tangent sheaves. Symmetric and generic determinants

“…This happens for instance for generic arrangements of at least N + 2 hyperplanes [DK93], but also for many highly non-generic arrangements cf. [FMV13,AFV16]. The stability of T D for hypersurfaces with isolated singularities was studied in [Dim17], in connection with the Torelli problem, on whether D can be reconstructed from T D .…”