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Arrangements of hyperplanes and vector bundles on $$\mathbb{P}^ n$$. (English) Zbl 0804.14007
Let $${\mathcal H} = \{H_ 1, \dots, H_ m\}$$ be an arrangement of $$m$$ hyperplanes in a $$n$$-dimensional projective space $$\mathbb{P}^ n$$. The authors consider the divisor $$D = H_ 1 \cup \cdots \cup H_ m$$ and the locally free sheaf (or bundle) $$\Omega^ 1 (\log D)$$ of the differential 1-forms with logarithmic poles on $$D$$. Main result: If $$m \geq 2n + 3$$, then the arrangement of hyperplanes $${\mathcal H}$$ can be recovered from the sheaf $$\Omega^ 1 (\log D)$$ unless all hyperplanes are osculating the same rational normal curve of degree $$n$$.
In case $$n=2$$ the variety of jumping lines of $$\Omega^ 1 (\log D)$$ is a curve in the dual $$\mathbb{P}^ 2$$ and the whole construction provides an interesting procedure that associates to a finite set of points in a plane a well determined curve through them.

##### MSC:
 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 14N10 Enumerative problems (combinatorial problems) in algebraic geometry
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##### References:
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