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Singular loci of reflection arrangements and the containment problem

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Abstract

This paper provides insights into the role of symmetry in studying polynomial functions vanishing to high order on an algebraic variety. The varieties we study are singular loci of hyperplane arrangements in projective space, with emphasis on arrangements arising from complex reflection groups. We provide minimal sets of equations for the radical ideals defining these singular loci and study containments between the ordinary and symbolic powers of these ideals. Our work ties together and generalizes results in Bauer et al. (Int Math Res Not IMRN 24:7459–7514, 2019), Dumnicki et al. (J Algebra 393:24–29, 2013), Harbourne and Seceleanu (J Pure Appl Algebra 219(4):1062–1072, 2015) and Malara and Szpond (J Pure Appl Algebra 222(8):2323–2329, 2018) under a unified approach.

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Acknowledgements

We thank Eloísa Grifo, Jack Jeffries and Tomasz Szemberg for useful comments. We acknowledge Thai Nguy\(\tilde{\hat{\text {e}}}\)n for pointing out an error in a previous version of the proof of Proposition 6.3. The authors acknowledge the support of NSF grant DMS-1601024 and EpSCOR award OIA-1557417 throughout the completion of this work.

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  1. Department of Mathematics, University of Nebraska-Lincoln, 203 Avery Hall, Lincoln, NE, 68588, USA

    Benjamin Drabkin & Alexandra Seceleanu

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  1. Benjamin Drabkin
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  2. Alexandra Seceleanu
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Correspondence to Alexandra Seceleanu.

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Drabkin, B., Seceleanu, A. Singular loci of reflection arrangements and the containment problem. Math. Z. 299, 867–895 (2021). https://doi.org/10.1007/s00209-021-02701-1

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