Freeness and multirestriction of hyperplane arrangements
type: Item
instance of: scholarly article
title: Freeness and multirestriction of hyperplane arrangements
zbMATH Open document ID: 1261.14028
DOI: 10.1112/S0010437X12000164
author: Mathias Schulze
published in: Compositio Mathematica
publication date: 2012-06-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1003.0917
Mathematics Subject Classification ID: 14N20
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 16W25
zbMATH DE Number: 6051026
zbMATH Keywords: hyperplane arrangement
zbMATH Keywords: free divisor
MaRDI profile type: Publication
arXiv ID: 1003.0917
cites work: Combinatorial construction of logarithmic differential forms
cites work: ON THE FREENESS OF 3-ARRANGEMENTS
cites work: Characterization of a free arrangement and conjecture of Edelman and Reiner
cites work: Generalized exponents of a free arrangement of hyperplanes and Shepherd- Todd-Brieskorn formula
cites work: Combinatorics and topology of complements of hyperplanes
cites work: The module of logarithmic \(p\)-forms of a locally free arrangement
cites work: The characteristic polynomial of a multiarrangement
cites work: A formula for the characteristic polynomial of an arrangement
OpenAlex ID: W3100461121
review text: A central hyperplane arrangement \({\mathcal A}\) in a vector space \(V\) of dimension \(\ell\) is said to be free if the module \(\Omega^1({\mathcal A})\) of logarithmic differential \(1\)-forms associated with is free. Terao's conjecture is that freeness is a combinatorial property (of simple arrangements). \(\mathcal A\) is called weakly tame if \(\Omega^1({\mathcal A})\) has projective dimension at most~\(1\). The result here is that, if \({\mathcal A}\) is weakly tame (or weakly dually tame, which is the same condition for vector fields instead of differentials), or if \(\ell\leq 4\), then \({\mathcal A}\) is free if its restriction to a hyperplane \(H\) is free and the characteristic polynomial restricts in the expected way.NEWLINENEWLINEThe converse to this is known (for all \(\ell\geq 3\)) and the result here was proved for \(\ell=3\) by \textit{M. Yoshinaga} [Bull. Lond. Math. Soc. 37, No. 1, 126--134 (2005; Zbl 1071.52019)]. The further progress comes from applying other results of Yoshinaga and from the crucial observation that the logarithmic differentials with poles along \(H\) split off from \(\Omega^1({\mathcal A})\): that is, if \(H=\{\alpha_H=0\}\) and \(S=K[V]\) then NEWLINE\[NEWLINE \Omega^1({\mathcal A})\cong S{{d\alpha_H}\over{\alpha_H}}\oplus {{d\alpha_H}\over{\alpha_H}}\wedge\Omega^1({\mathcal A}). NEWLINE\]NEWLINE This is described by the author as ``trivial'' but it is what gives the condition of tameness its good inductive properties.NEWLINENEWLINEThe condition on the characteristic polynomial can be interpreted more geometrically in terms of characteristic polynomials of non-central arrangements (the intersections of \({\mathcal A}\) with \(\{\alpha_H=\epsilon\}\)) and central multiarrangements (i.e.\ with multiplicities) such as the restriction of \({\mathcal A}\) to \(H\). However, there does not seem to be a satisfactory way to treat characteristic polynomials for both these cases together. Moreover, the multiplicities in multiarrangeme…
reviewed by: G. K. Sankaran
description: scientific article; zbMATH DE number 6051026