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Critical Points and Resonance of Hyperplane Arrangements |
Canad. J. Math. 63(2011), 1038-1057
Published:2011-04-30
Printed: Oct 2011
Printed: Oct 2011
- D. Cohen,
- G. Denham,
- M. Falk,
- A. Varchenko,
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] |