An m-free hyperplane arrangement is a generalization of a free arrangement. Holm asked the following two questions: (1)Does m-free imply (m + 1)-free for any arrangement? (2)Are all arrangements m-free for m large enough? In this paper, we characterize m-freeness for product arrangements, while we prove that all localizations of an m-free arrangement are m-free. From these results, we give answers to Holm’s questions.

中文翻译：

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产品安排的高阶自由度的表征以及对霍尔姆问题的解答

一米-free超平面布置是自由布置的概括。Holm提出了以下两个问题：（1）m -free是否意味着（m +1）-free对任何安排都没有？（2）均安排中号-免费的米够大？在本文中，我们表征米-freeness产品的安排，而我们证明了的所有本地化米-免费安排是米-免费。从这些结果中，我们给出了霍尔姆问题的答案。