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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.

A Characterization of High Order Freeness for Product Arrangements and Answers to Holm’s Questions