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

A. Varchenko

Cohen

D. Cohen

Damon

Denham

Eisenbud

Esnault

Falk

Farber

G. Denham

Grayson

M. Falk

Musta

Orlik

Reshetikhin

Schechtman

Scherbak

Terao

Varchenko

English

arXiv

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

Cohen, D.

Denham, G.

Falk, M.

Varchenko, A.

Carolina Digital Repository

Critical Points and Resonance of Hyperplane Arrangements

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. © Canadian Mathematical Society 2011

LSU Scholarly Repository (Louisiana State Univ.)

Critical points and resonance of hyperplane arrangements

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

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Canadian Journal of Mathematics

https://repository.lsu.edu/cgi/viewcontent.cgi?article=2453&context=mathematics_pubs

Critical points and resonance of hyperplane arrangements

Abstract

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LSU Scholarly Repository (Louisiana State Univ.)

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