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ON THE INTERSECTION OF RATIONAL TRANSVERSAL SUBTORI

Part of: (Co)homology theory Singularities Cycles and subschemes Curves

Published online by Cambridge University Press:  01 April 2009

SHAHEEN NAZIR
SHAHEEN NAZIR*
Affiliation:
Abdus Salam School of Mathematical Sciences, Government College University, Lahore, Pakistan (email: shaheen.nazeer@gmail.com)
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Abstract

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We show that under a suitable transversality condition, the intersection of two rational subtori in an algebraic torus (ℂ*)n is a finite group which can be determined using the torsion part of some associated lattice. We also give applications to the study of characteristic varieties of smooth complex algebraic varieties. As an example we discuss A. Suciu’s line arrangement, the so-called deleted B3-arrangement.

Keywords

local systemconstructible sheaftwisted cohomologycharacteristic varietypencil of plane curves

MSC classification

Secondary: 14C21: Pencils, nets, webs 14F99: None of the above, but in this section 32S22: Relations with arrangements of hyperplanes 14H50: Plane and space curves
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 86 , Issue 2 , April 2009 , pp. 221 - 231
Copyright
Copyright © Australian Mathematical Society 2009

References

[1]Arapura, D., ‘Geometry of cohomology support loci for local systems. I’, J. Algebraic Geom. 6 (1997), 563–597.Google Scholar
[2]Beauville, A., ‘Annulation du H 1 pour les fibrés en droites plats’, in: Complex Algebraic Varieties (Bayreuth, 1990), Lecture Notes in Mathematics, 1507 (Springer, Berlin, 1992), pp. 1–15.Google Scholar
[3]Cohen, D. C. and Orlik, P., ‘Arrangements and local systems’, Math. Res. Lett. 7 (2000), 299–316.CrossRefGoogle Scholar
[4]Cohen, D. C. and Suciu, A. I., ‘Characteristic varieties of arrangements’, Math. Proc. Cambridge Philos. Soc. 127 (1999), 33–54.CrossRefGoogle Scholar
[5]Dimca, A., ‘Pencils of plane curves and characteristic varieties’, Preprint, 2006, math/0606442.Google Scholar
[6]Dimca, A., Papadima, S. and Suciu, A., ‘Formality, Alexander invariants, and a question of Serre’, Preprint, 2005, math/0512480.Google Scholar
[7]Dimca, A., Papadima, S. and Suciu, A., ‘Alexander polynomials: essential variables and multiplicities’, Int. Math. Res. Not. 2008 (2008), 36 pages, doi:10.1093/imrn/rnm119.Google Scholar
[8]Esnault, H., Schechtman, V. and Viehweg, E., ‘Cohomology of local systems on the complement of hyperplanes’, Invent. Math. 109 (1992), 557–561. Erratum, Invent. Math. 112 (1993) 447.CrossRefGoogle Scholar
[9]Libgober, A., ‘First order deformations for rank one local systems with a non-vanishing cohomology’, Topology Appl. 118(1–2) (2002), 159–168.CrossRefGoogle Scholar
[10]Nazir, S., ‘Hyperplane arrangements’, PhD Thesis, Abdus Salam School of Mathematical Sciences, GC University Lahore, 2007.Google Scholar
[11]Orlik, P. and Terao, H., Arrangements of Hyperplanes (Springer, Berlin, 1992).CrossRefGoogle Scholar
[12]Simpson, C., ‘Subspaces of moduli spaces of rank one local systems’, Ann. Sci. École Norm. Sup. 26(3) (1993), 361–401.CrossRefGoogle Scholar
[13]Suciu, A., ‘Translated tori in the characteristic varieties of complex hyperplane arrangements’, Topology Appl. 118(1–2) (2002), 209–223.CrossRefGoogle Scholar
[14]Suciu, A., ‘Fundamental groups of line arrangements: enumerative aspects’, in: Advances in Algebraic Geometry Motivated by Physics (Lowell, MA, 2000), Contemporary Mathematics, 276 (American Mathematical Society, Providence, RI, 2001), pp. 43–79.CrossRefGoogle Scholar