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2011•Canadian Journal of MathematicsOpen access

Critical Points and Resonance of Hyperplane Arrangements

Daniel C. Cohen, Graham Denham, Michael Falk, Alexander Varchenko

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Abstract

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.

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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.

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Available abstract

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.

Key concepts: Hyperplane, Mathematics, Codimension, Rank (graph theory), Dimension (graph theory), Combinatorics, Variety (cybernetics), Resonance (particle physics)

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