Abstract
The class of free multiarrangements is known to be closed under taking localizations. We extend this result to the stronger notions of inductive and recursive freeness. As an application, we prove that recursively free (multi)arrangements are compatible with the product construction for (multi)arrangements. In addition, we show how our results can be used to derive that some canonical classes of free multiarrangements are not inductively free.
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T. Abe, K. Nuida, Y. Numata, Signed-eliminable graphs and free multiplicities on the braid arrangement. J. Lond. Math. Soc. (2) 80(1), 121–134 (2009)
T. Abe, H. Terao, M. Wakefield, The Euler multiplicity and addition-deletion theorems for multiarrangements. J. Lond. Math. Soc. (2) 77(2), 335–348 (2008)
T. Abe, M. Yoshinaga, Free arrangements and coefficients of characteristic polynomials. Math. Z. 275(3–4), 911–919 (2013)
N. Amend, T. Hoge, G. Röhrle,On inductively free restrictions of reflection arrangements. J. Algebra 418, 197–212 (2014)
M. Barakat, M. Cuntz, Coxeter and crystallographic arrangements are inductively free. Adv. Math. 229, 691–709 (2012)
M. Cuntz, T. Hoge, Free but not recursively free arrangements. Proc. Am. Math. Soc. 143, 35–40 (2015)
T. Hoge, G. Röhrle, On inductively free reflection arrangements. J. Reine Angew. Math. 701, 205–220 (2015)
P. Mücksch, On recursively free reflection arrangements. J. Algebra 474, 24–48 (2017)
P. Orlik, L. Solomon, Arrangements defined by unitary reflection groups. Math. Ann. 261, 339–357 (1982)
P. Orlik, H. Terao, Arrangements of Hyperplanes (Springer, Berlin, 1992)
M. Schulze, Freeness and multirestriction of hyperplane arrangements. Compos. Math. 148(3), 799–806 (2012)
H. Terao, Arrangements of hyperplanes and their freeness I, II. J. Fac. Sci. Univ. Tokyo 27, 293–320 (1980)
H. Terao, Generalized exponents of a free arrangement of hyperplanes and Shepherd-Todd-Brieskorn formula. Invent. Math. 63(1), 159–179 (1981)
H. Terao, Free arrangements of hyperplanes over an arbitrary field. Proc. Jpn. Acad. Ser. A Math. Sci. 59(7), 301–303 (1983)
M. Yoshinaga, Characterization of a free arrangement and conjecture of Edelman and Reiner. Invent. Math. 157(2), 449–454 (2004)
M. Yoshinaga, On the freeness of 3-arrangements. Bull. Lond. Math. Soc. 37(1), 126–134 (2005)
M. Yoshinaga, Freeness of hyperplane arrangements and related topics. Ann. Fac. Sci. Toulouse Math. (6) 23(2), 483–512 (2014)
G. Ziegler, Multiarrangements of hyperplanes and their freeness, in Singularities (Iowa City, IA, 1986). Contemporary Mathematics, vol. 90 (American Mathematical Society, Providence, RI, 1989), pp. 345–359
G. Ziegler, Matriod representations and free arrangements. Trans. Am. Math. Soc. 320, 525–541 (1990)
Acknowledgements
We acknowledge support from the DFG-priority program SPP1489 “Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory”.
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Hoge, T., Röhrle, G., Schauenburg, A. (2017). Inductive and Recursive Freeness of Localizations of Multiarrangements. In: Böckle, G., Decker, W., Malle, G. (eds) Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-70566-8_16
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DOI: https://doi.org/10.1007/978-3-319-70566-8_16
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