TY - JOUR
T1 - Locally heavy hyperplanes in multiarrangements
AU - Abe, Takuro
AU - Kühne, Lukas
N1 - Funding Information:
We thank the anonymous referee for carefully reading an earlier version of this article and for giving many useful suggestions how to improve the exposition. The first author is partially supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (B) 16H03924. The second author is supported by ERC StG 716424 - CASe and a Minerva Fellowship of the Max Planck Society.
Publisher Copyright:
© 2021 Elsevier B.V.
PY - 2022/1
Y1 - 2022/1
N2 - Hyperplane arrangements of rank 3 admitting an unbalanced Ziegler restriction are known to fulfill Terao's conjecture. This long-standing conjecture asks whether the freeness of an arrangement is determined by its combinatorics. In this note we prove that arrangements which admit a locally heavy flag satisfy Terao's conjecture which is a generalization of the statement above to arbitrary dimension. To this end we extend results characterizing the freeness of multiarrangements with a heavy hyperplane to those satisfying the weaker notion of a locally heavy hyperplane. As a corollary we give a new proof that irreducible arrangements with a generic hyperplane are totally nonfree. In another application we show that an irreducible multiarrangement of rank 3 with at least two locally heavy hyperplanes is not free.
AB - Hyperplane arrangements of rank 3 admitting an unbalanced Ziegler restriction are known to fulfill Terao's conjecture. This long-standing conjecture asks whether the freeness of an arrangement is determined by its combinatorics. In this note we prove that arrangements which admit a locally heavy flag satisfy Terao's conjecture which is a generalization of the statement above to arbitrary dimension. To this end we extend results characterizing the freeness of multiarrangements with a heavy hyperplane to those satisfying the weaker notion of a locally heavy hyperplane. As a corollary we give a new proof that irreducible arrangements with a generic hyperplane are totally nonfree. In another application we show that an irreducible multiarrangement of rank 3 with at least two locally heavy hyperplanes is not free.
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U2 - 10.1016/j.jpaa.2021.106791
DO - 10.1016/j.jpaa.2021.106791
M3 - Article
AN - SCOPUS:85106942295
VL - 226
JO - Journal of Pure and Applied Algebra
JF - Journal of Pure and Applied Algebra
SN - 0022-4049
IS - 1
M1 - 106791
ER -