Critical points and resonance of hyperplane arrangements

D. Cohen; G. Denham; M. Falk; A. Varchenko

doi:10.4153/CJM-2011-028-8

Critical points and resonance of hyperplane arrangements

D. Cohen, G. Denham, M. Falk, A. Varchenko

Mathematics and Statistics

Research output: Contribution to journal › Article › peer-review

15 Scopus citations

Abstract

If Φ_λ is a master function corresponding to a hyperplane arrangement A and a collection of weights λ, we investigate the relationship between the critical set of Φ_λ, the variety defined by the vanishing of the one-form ωλ = d log and Φ_λ, the resonance of λ. For arrangements satisfying certain conditions, we show that if λ is resonant in dimension p, then the critical set of Φ_λ has codimension at most p. These include all free arrangements and all rank 3 arrangements.

Original language	English (US)
Pages (from-to)	1038-1057
Number of pages	20
Journal	Canadian Journal of Mathematics
Volume	63
Issue number	5
DOIs	https://doi.org/10.4153/CJM-2011-028-8
State	Published - Oct 2011

Keywords

Critical set
Hyperplane arrangement
Master function
Resonant weights

ASJC Scopus subject areas

General Mathematics

Access to Document

10.4153/CJM-2011-028-8

Cite this

@article{85a29f748561426985c607be9674e160,

title = "Critical points and resonance of hyperplane arrangements",

abstract = "If Φλ is a master function corresponding to a hyperplane arrangement A and a collection of weights λ, we investigate the relationship between the critical set of Φλ, the variety defined by the vanishing of the one-form ωλ = d log and Φλ, the resonance of λ. For arrangements satisfying certain conditions, we show that if λ is resonant in dimension p, then the critical set of Φλ has codimension at most p. These include all free arrangements and all rank 3 arrangements.",

keywords = "Critical set, Hyperplane arrangement, Master function, Resonant weights",

author = "D. Cohen and G. Denham and M. Falk and A. Varchenko",

year = "2011",

month = oct,

doi = "10.4153/CJM-2011-028-8",

language = "English (US)",

volume = "63",

pages = "1038--1057",

journal = "Canadian Journal of Mathematics",

issn = "0008-414X",

publisher = "Canadian Mathematical Society",

number = "5",

}

TY - JOUR

T1 - Critical points and resonance of hyperplane arrangements

AU - Cohen, D.

AU - Denham, G.

AU - Falk, M.

AU - Varchenko, A.

PY - 2011/10

Y1 - 2011/10

N2 - If Φλ is a master function corresponding to a hyperplane arrangement A and a collection of weights λ, we investigate the relationship between the critical set of Φλ, the variety defined by the vanishing of the one-form ωλ = d log and Φλ, the resonance of λ. For arrangements satisfying certain conditions, we show that if λ is resonant in dimension p, then the critical set of Φλ has codimension at most p. These include all free arrangements and all rank 3 arrangements.

AB - If Φλ is a master function corresponding to a hyperplane arrangement A and a collection of weights λ, we investigate the relationship between the critical set of Φλ, the variety defined by the vanishing of the one-form ωλ = d log and Φλ, the resonance of λ. For arrangements satisfying certain conditions, we show that if λ is resonant in dimension p, then the critical set of Φλ has codimension at most p. These include all free arrangements and all rank 3 arrangements.

KW - Critical set

KW - Hyperplane arrangement

KW - Master function

KW - Resonant weights

UR - http://www.scopus.com/inward/record.url?scp=84861996446&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84861996446&partnerID=8YFLogxK

U2 - 10.4153/CJM-2011-028-8

DO - 10.4153/CJM-2011-028-8

M3 - Article

AN - SCOPUS:84861996446

SN - 0008-414X

VL - 63

SP - 1038

EP - 1057

JO - Canadian Journal of Mathematics

JF - Canadian Journal of Mathematics

IS - 5

ER -

Critical points and resonance of hyperplane arrangements

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this