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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Admissible local systems for a class of line arrangements


Authors: Shaheen Nazir and Zahid Raza
Journal: Proc. Amer. Math. Soc. 137 (2009), 1307-1313
MSC (2000): Primary 14C21, 14F99, 32S22; Secondary 14E05, 14H50.
Published electronically: November 6, 2008
MathSciNet review: 2465653
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Abstract: A rank one local system $ \mathcal{L}$ on a smooth complex algebraic variety $ M$ is admissible if roughly speaking the dimension of the cohomology groups $ H^m(M,\mathcal{L})$ can be computed directly from the cohomology algebra $ H^{\*}(M,\mathbb{C})$.

We say that a line arrangement $ \mathcal{A}$ is of type $ \mathcal{C}_k$ for some $ k\ge 0 $ if $ k$ is the minimal number of lines in $ \mathcal{A}$ containing all the points of multiplicity at least 3. We show that if $ \mathcal{A}$ is a line arrangement in the classes $ \mathcal{C}_k$ for $ k\leq 2$, then any rank one local system $ \mathcal{L}$ on the line arrangement complement $ M$ is admissible. Partial results are obtained for the class $ \mathcal{C}_3$.


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Additional Information

Shaheen Nazir
Affiliation: Abdus Salam School of Mathematical Sciences, Government College University,68-B New Muslim Town, Lahore, Pakistan
Address at time of publication: Abdus Salam School of Mathematical Sciences, Government College University, 35 C-2 Gulberg III, Lahore, Pakistan
Email: shaheen.nazeer@gmail.com

Zahid Raza
Affiliation: Abdus Salam School of Mathematical Sciences, Government College University, 68-B New Muslim Town, Lahore, Pakistan
Email: zahidsms@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09661-5
PII: S 0002-9939(08)09661-5
Keywords: Admissible local system, line arrangement, characteristic variety
Received by editor(s): January 22, 2008
Received by editor(s) in revised form: June 2, 2008
Published electronically: November 6, 2008
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.