Abstract
We prove the addition–deletion theorems for the Solomon–Terao polynomials, which have two important specializations. Namely, one is to the characteristic polynomials of hyperplane arrangements, and the other to the Poincarè polynomials of the regular nilpotent Hessenberg varieties. One of the main tools to show them is the free surjection theorem which confirms the right exactness of several important exact sequences among logarithmic modules. Moreover, we introduce a generalized polynomial B-theory to the higher order logarithmic modules, whose origin was due to Terao.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
References
Abe, T.: Divisionally free arrangements of hyperplanes. Invent. Math. 204(1), 317–346 (2016)
Abe, T.: Plus-one generated and next to free arrangements of hyperplanes. Int. Math. Res. Not. 2021(12), 9233–9261 (2021)
Abe, T.: Double points of free projective line arrangements. Int. Math. Res. Not. 2022(3), 1811–1824 (2022)
Abe, T.: Addition–deletion theorem for free hyperplane arrangements and combinatorics. J. Algebra 610, 1–17 (2022)
Abe, T.: Projective dimensions of hyperplane arrangements (2020). arXiv:2009:04101
Abe, T., Denham, G.: Deletion-restriction for logarithmic forms on multiarrangements (2022). arXiv:2203.04816
Abe, T., Horiguchi, T., Masuda, M., Murai, S., Sato, T.: Hessenberg varieties and hyperplane arrangements. J. Reine Angew. Math. 764, 241–286 (2020)
Abe, T., Maeno, T., Murai, S., Numata, Y.: Solomon–Terao algebra for hyperplane arrangements. J. Math. Soc. Jpn. 71(4), 1027–1047 (2019)
Kung, J., Schenck, H.: Derivation modules of orthogonal duals of hyperplane arrangements. J. Algebraic Comb. 24(3), 253–262 (2006)
Mustaţă, M., Schenck, H.: The module of logarithmic \(p\)-forms of a locally free arrangement. J. Algebra 241(2), 699–719 (2001)
Orlik, P., Solomon, L.: Combinatorics and topology of complements of hyperplanes. Invent. Math. 56(2), 167–189 (1980)
Orlik, P., Terao, H.: Arrangements of Hyperplanes. Grundlehren der Mathematischen Wissenschaften, vol. 300. Springer, Berlin (1992)
Schenck, H., Terao, H., Yoshinaga, M.: Logarithmic vector fields for curve configurations in \({\textbf{P} }^{2}\) with quasihomogeneous singularities. Math. Res. Lett. 25(6), 1977–1992 (2018)
Solomon, L., Terao, H.: A formula for the characteristic polynomial of an arrangement. Adv. Math. 64, 305–325 (1987)
Terao, H.: Arrangements of hyperplanes and their freeness I, II. J. Fac. Sci. Univ. Tokyo 27, 293–320 (1980)
Ziegler, G.M.: Combinatorial construction of logarithmic differential forms. Adv. Math. 76, 116–154 (1989)
Acknowledgements
The author is partially supported by JSPS KAKENHI Grant numbers JP18KK0389 and JP21H00975.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Abe, T. Addition–deletion theorems for the Solomon–Terao polynomials and B-sequences of hyperplane arrangements. Math. Z. 306, 25 (2024). https://doi.org/10.1007/s00209-023-03426-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00209-023-03426-z