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arXiv · 2112.13462

Geometry of logarithmic derivations of hyperplane arrangements

Avi Steiner·Graham Denham
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Abstract

We study the Hadamard product of the linear forms defining a hyperplane arrangement with those of its dual, which we view as generating an ideal in a certain polynomial ring. We use this ideal, which we call the ideal of pairs, to study logarithmic derivations and critical set varieties of arrangements in a way which is symmetric with respect to matroid duality. Our main result exhibits the variety of the ideal of pairs as a subspace arrangement whose components correspond to cyclic flats of the arrangement. As a corollary, we are able to give geometric explanations of some freeness and projective dimension results due to Ziegler and Kung--Schenck.

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Avi Steiner, Graham Denham. 2021-12-26. Geometry of logarithmic derivations of hyperplane arrangements. https://arxiv.org/abs/2112.13462

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