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I'm working on a problem related to $\textbf{Randell's isotopy theorem}$ for complex hyperplane arrangements. I have a question which seems quite obvious. However, I haven't found a rigorous proof yet. I'm pretty sure this is a well known fact in hyperplane arrangements theory. All rigorous definitions and notations are available in Randell's original paper and in this expository notes by Stanley.

$\textbf{Question}$ Let $\mathcal{A}$ and $\mathcal{B}$ two complex hyperplanes arrangements in $\mathbb{C}^{d}.$ If the essentializations $\operatorname{ess}(\mathcal{A})$ and $\operatorname{ess}(\mathcal{B})$ are lattice-isotopic, then the arrangements $\mathcal{A}$ and $\mathcal{B}$ are lattice-isotopic.

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  • $\begingroup$ Does the obvious proof (explicitly construct the isotopy) not work? $\endgroup$ Apr 29, 2015 at 20:13
  • $\begingroup$ You are right. I only had some problems in writing down the explicit construction of the lattice isotopy from the one at the level of essential arrangements. Fortunately, I was able to solve. $\endgroup$
    – snaleimath
    Apr 30, 2015 at 8:28

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