Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Torelli problem for Logarithmic bundles of hypersurface arrangements
  in the projective space   Let $ \mathcal{D} = \{D_{1}, \ldots, D_{\ell}\} $ be an arrangement of smooth
hypersurfaces with normal crossings on the complex projective space $
\mathbb{P}^{n} $ and let $ \Omega^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $ be
the logarithmic bundle attached to it. Our aim is to study the injectivity of
the correspondence $ \mathcal{D} \longrightarrow
\Omega^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $. In order to do that, we first
show that $ \Omega^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $ admits a resolution
of length $ 1 $ depending on the degrees and on the equations of $ D_{1},
\ldots, D_{\ell} $. Then, we prove a Torelli type theorem when $ \mathcal{D} $
has a sufficiently large number of components of the same degree $ d $, by
recovering them as unstable smooth irreducible degree-$d$ hypersurfaces of $
\Omega^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $. The cases of one quadric and a
pair of quadrics in $ \mathbb{P}^{n} $ are not Torelli; in particular, through
a duality argument, we prove that the isomorphism class of the logarithmic
bundle attached to a pair of quadrics is determined by the tangent hyperplanes
to the pair. Finally, by describing the moduli spaces containing $
\Omega^{1}_{\mathbb{P}^{2}}(log \mathcal{D}) $, we show that some line-conic
arrangements are not of Torelli type.

24
The categories of this paper are: dg-ga
14
The categories of this paper are: math-ph
23
The categories of this paper are: solv-int
22
The categories of this paper are: chao-dyn
31
The categories of this paper are: ao-sci
36
The categories of this paper are: cmp-lg
32
The categories of this paper are: atom-ph
19
The categories of this paper are: eess
27
The categories of this paper are: comp-gas
13
The categories of this paper are: quant-ph
3
The categories of this paper are: physics
16
The categories of this paper are: stat
28
The categories of this paper are: alg-geom
35
The categories of this paper are: mtrl-th
18
The categories of this paper are: econ
7
The categories of this paper are: hep-th