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# Admissibility of Local Systems for some Classes of Line Arrangements

Published:2014-07-10
Printed: Sep 2014
• Nguyen Tat Thang,
Institute of Mathematics, Vietnam Academy of Science and Technology, 10307 Hanoi, Vietnam
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## Abstract

Let $\mathcal{A}$ be a line arrangement in the complex projective plane $\mathbb{P}^2$ and let $M$ be its complement. A rank one local system $\mathcal{L}$ on $M$ is admissible if roughly speaking the cohomology groups $H^m(M,\mathcal{L})$ can be computed directly from the cohomology algebra $H^{*}(M,\mathbb{C})$. In this work, we give a sufficient condition for the admissibility of all rank one local systems on $M$. As a result, we obtain some properties of the characteristic variety $\mathcal{V}_1(M)$ and the Resonance variety $\mathcal{R}_1(M)$.
 Keywords: admissible local system, line arrangement, characteristic variety, multinet, resonance variety
 MSC Classifications: 14F99 - None of the above, but in this section 32S22 - Relations with arrangements of hyperplanes [See also 52C35] 52C35 - Arrangements of points, flats, hyperplanes [See also 32S22] 05A18 - Partitions of sets 05C40 - Connectivity 14H50 - Plane and space curves