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Freeness and multirestriction of hyperplane arrangements

Part of: Projective and enumerative geometry Commutative algebra: Homological methods Rings and algebras with additional structure

Published online by Cambridge University Press:  22 March 2012

Mathias Schulze
Mathias Schulze*
Affiliation:
Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA (email: mschulze@math.okstate.edu)
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Abstract

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Generalizing a result of Yoshinaga in dimension three, we show that a central hyperplane arrangement in 4-space is free exactly if its restriction with multiplicities to a fixed hyperplane of the arrangement is free and its reduced characteristic polynomial equals the characteristic polynomial of this restriction. We show that the same statement holds true in any dimension when imposing certain tameness hypotheses.

Keywords

hyperplane arrangementfree divisor

MSC classification

Secondary: 14N20: Configurations and arrangements of linear subspaces 16W25: Derivations, actions of Lie algebras 13D40: Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 3 , May 2012 , pp. 799 - 806
Copyright
Copyright © Foundation Compositio Mathematica 2012

References

[ATW07]Abe, T., Terao, H. and Wakefield, M., The characteristic polynomial of a multiarrangement, Adv. Math. 215 (2007), 825–838; MR 2355609.CrossRefGoogle Scholar
[MS01]Mustaţǎ, M. and Schenck, H. K., The module of logarithmic p-forms of a locally free arrangement, J. Algebra 241 (2001), 699–719; MR 1843320(2002c:32047).CrossRefGoogle Scholar
[OS80]Orlik, P. and Solomon, L., Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), 167–189; MR 558866(81e:32015).Google Scholar
[OT92]Orlik, P. and Terao, H., Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 300 (Springer, Berlin, 1992); MR 1217488(94e:52014).CrossRefGoogle Scholar
[ST87]Solomon, L. and Terao, H., A formula for the characteristic polynomial of an arrangement, Adv. Math. 64 (1987), 159–179.Google Scholar
[Ter81]Terao, H., Generalized exponents of a free arrangement of hyperplanes and Shepherd-Todd-Brieskorn formula, Invent. Math. 63 (1981), 159–179; MR 608532(82e:32018b).CrossRefGoogle Scholar
[Yos04]Yoshinaga, M., Characterization of a free arrangement and conjecture of Edelman and Reiner, Invent. Math. 157 (2004), 449–454; MR 2077250(2005d:52044).CrossRefGoogle Scholar
[Yos05]Yoshinaga, M., On the freeness of 3-arrangements, Bull. Lond. Math. Soc. 37 (2005), 126–134; MR 2105827(2005i:52030).CrossRefGoogle Scholar
[Zie89a]Ziegler, G. M., Combinatorial construction of logarithmic differential forms, Adv. Math. 76 (1989), 116–154; MR 1004488(90j:32016).CrossRefGoogle Scholar
[Zie89b]Ziegler, G. M., Multiarrangements of hyperplanes and their freeness, in Singularities (Iowa City, IA, 1986), Contemporary Mathematics, vol. 90 (American Mathematical Society, Providence, RI, 1989), 345–359; MR 1000610(90e:32015).Google Scholar