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Critical Points and Resonance of Hyperplane Arrangements
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- Journal:
- Canadian Journal of Mathematics / Volume 63 / Issue 5 / 18 October 2011
- Published online by Cambridge University Press:
- 20 November 2018, pp. 1038-1057
- Print publication:
- 18 October 2011
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- Article
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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.
