Abstract
We show that the deletion theorem of a free arrangement is combinatorial, i.e., whether we can delete a hyperplane from a free arrangement keeping freeness depends only on the intersection lattice. In fact, we give a sufficient and necessary condition for the deletion theorem in terms of characteristic polynomials. As a corollary, we prove that whether a free arrangement has a free filtration is also combinatorial. The proof is based on the result about a minimal set of generators of a logarithmic derivation module of a multiarrangement which satisfies the \(b_2\)-equality.
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References
Abe, T.: Roots of characteristic polynomials and and intersection points of line arrangements. J. Singularities 8, 100–117 (2014)
Abe, T.: Divisionally free arrangements of hyperplanes. Invent. Math. 204(1), 317–346 (2016)
Abe, T.: Restrictions of Free Arrangements and the Division Theorem. In: Callegaro, F., Carnovale, G., Caselli, F., De Concini, C., De Sole, A. (eds.) Perspectives in Lie Theory. Springer INdAM Series, vol. 19, pp. 389–401. Springer, Cham (2017)
Abe, T., Terao, H.: Free filtrations of affine Weyl arrangements and the ideal-Shi arrangements. J. Alg. Combin. 43(1), 33–44 (2016)
Abe, T., Terao, H., Wakefield, M.: The characteristic polynomial of a multiarrangement. Adv. in Math. 215, 825–838 (2007)
Abe, T., Terao, H., Wakefield, M.: The Euler multiplicity and addition-deletion theorems for multiarrangements. J. London Math. Soc. 77(2), 335–348 (2008)
Abe, T., Yoshinaga, M.: Free arrangements and coefficients of characteristic polynomials. Math. Z. 275(3), 911–919 (2013)
Dimca, A., Sticlaru, G.: Nearly free divisors and rational cuspidal curves. arXiv:1505.00666
Dimca, A., Sticlaru, G.: Free and nearly free surfaces in \(\mathbf{P}^3\), to appear in Asian J. Math., arXiv:1507.03450
Edelman, P.H., Reiner, V.: A counterexample to Orlik’s conjecture. Proc. Amer. Math. Soc. 118, 927–929 (1993)
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, p. 300. Springer, Berlin (1992)
Rybnikov, G.L.: On the fundamental group of the complement of a complex hyperplane arrangement. Funktsional. Anal. i Prilozhen. 45(2), 71–85 (2011)
Saito, K.: Theory of logarithmic differential forms and logarithmic vector fields. J. Fac. Sci. Univ. Tokyo 27, 265–291 (1980)
Terao, H.: Arrangements of hyperplanes and their freeness I. II. J. Fac. Sci. Univ. Tokyo 27, 293–320 (1980)
Terao, H.: Generalized exponents of a free arrangement of hyperplanes and Shephard-Todd-Brieskorn formula. Invent. math. 63, 159–179 (1981)
Yoshinaga, M.: On the freeness of 3-arrangements. Bull. London Math. Soc. 37(1), 126–134 (2005)
Yoshinaga, M.: Freeness of hyperplane arrangements and related topics. Anna. de la Faculte des Sci. de Toulouse 23(2), 483–512 (2014)
Ziegler, G. M.: Multiarrangements of hyperplanes and their freeness. Singularities (Iowa City, IA, 1986). In: Contemporary Mathematics, vol. 90, pp. 345–359. American Mathematical Society, Providence, RI (1989)
Acknowledgements
The author is grateful to the anonymous referees for several comments and suggestions to this article. The author is partially supported by JSPS Grant-in-Aid for Scientific Research (B) JP16H03924, and Grant-in-Aid for Exploratory Research JP16K13744. We are grateful to Michael DiPasquale for informing an example in Remark 3.6.
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Communicated by Thomas Schick.