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Critical Points and Resonance of Hyperplane Arrangements

Published online by Cambridge University Press:  20 November 2018

D. Cohen ,
G. Denham ,
M. Falk  and
A. Varchenko
D. Cohen
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A. email: cohen@math.lsu.edu
G. Denham
Affiliation:
Department of Mathematics, University of Western Ontario, London, ON N6A 5B7 email: gdenham@uwo.ca
M. Falk
Affiliation:
Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, AZ 86011, U.S.A. email: michael.falk@nau.edu
A. Varchenko
Affiliation:
Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, U.S.A. email: anv@email.unc.edu
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Abstract

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

Keywords

32S2255N2552C35hyperplane arrangementmaster functionresonant weightscritical set
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 63 , Issue 5 , 18 October 2011 , pp. 1038 - 1057
Copyright
Copyright © Canadian Mathematical Society 2011

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