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# Critical Points and Resonance of Hyperplane Arrangements

Published:2011-04-30
Printed: Oct 2011
• D. Cohen,
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A.
• G. Denham,
Department of Mathematics, University of Western Ontario, London, ON N6A 5B7
• M. Falk,
Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, AZ 86011, U.S.A.
• A. Varchenko,
Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, U.S.A.
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## Abstract

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 $\omega_\lambda=\operatorname{d} \log \Phi_\lambda$, 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: hyperplane arrangement, master function, resonant weights, critical set
 MSC Classifications: 32S22 - Relations with arrangements of hyperplanes [See also 52C35] 55N25 - Homology with local coefficients, equivariant cohomology 52C35 - Arrangements of points, flats, hyperplanes [See also 32S22]

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