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Euclidean matchings and minimality of hyperplane arrangements

Lofano, Davide and Paolini, Giovanni (2021) Euclidean matchings and minimality of hyperplane arrangements. Discrete Mathematics, 344 (3). Art. No. 112232. ISSN 0012-365X. https://resolver.caltech.edu/CaltechAUTHORS:20210211-131854902

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Abstract

We construct a new class of maximal acyclic matchings on the Salvetti complex of a locally finite hyperplane arrangement. Using discrete Morse theory, we then obtain an explicit proof of the minimality of the complement. Our construction provides interesting insights also in the well-studied case of finite arrangements, and gives a nice geometric description of the Betti numbers of the complement. In particular, we solve a conjecture of Drton and Klivans on the characteristic polynomial of finite reflection arrangements. The minimal complex is compatible with restrictions, and this allows us to prove the isomorphism of Brieskorn’s Lemma by a simple bijection of the critical cells. Finally, in the case of line arrangements, we describe the algebraic Morse complex which computes the homology with coefficients in an abelian local system.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1016/j.disc.2020.112232DOIArticle
https://arxiv.org/abs/1809.02476arXivDiscussion Paper
ORCID:
AuthorORCID
Lofano, Davide0000-0002-9478-316X
Paolini, Giovanni0000-0002-3964-9101
Additional Information:© 2020 Elsevier B.V. Received 17 March 2020, Accepted 7 November 2020, Available online 2 December 2020. We would like to thank our supervisor, Mario Salvetti, for always giving valuable advice. We also thank Emanuele Delucchi, Simona Settepanella, Federico Glaudo, and Daniele Semola, for the useful discussions. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Subject Keywords:Hyperplane arrangements; Discrete Morse theory
Issue or Number:3
Record Number:CaltechAUTHORS:20210211-131854902
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20210211-131854902
Official Citation:Davide Lofano, Giovanni Paolini, Euclidean matchings and minimality of hyperplane arrangements, Discrete Mathematics, Volume 344, Issue 3, 2021, 112232, ISSN 0012-365X, https://doi.org/10.1016/j.disc.2020.112232.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:108007
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:11 Feb 2021 21:37
Last Modified:11 Feb 2021 21:37

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