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Reducibility of hyperplane arrangements

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Abstract

Certain problems on reducibility of central hyperplane arrangements are settled. Firstly, a necessary and sufficient condition on reducibility is obtained. More precisely, it is proved that the number of irreducible components of a central hyperplane arrangement equals the dimension of the space consisting of the logarithmic derivations of the arrangement with degree zero or one. Secondly, it is proved that the decomposition of an arrangement into a direct sum of its irreducible components is unique up to an isomorphism of the ambient space. Thirdly, an effective algorithm for determining the number of irreducible components and decomposing an arrangement into a direct sum of its irreducible components is offered. This algorithm can decide whether an arrangement is reducible, and if it is the case, what the defining equations of irreducible components are.

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Authors and Affiliations

  1. Department of Mathematics, Beijing University of Chemical Technology, Beijing, 100029, China

    Guang-feng Jiang

  2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100080, China

    Jian-ming Yu

Authors
  1. Guang-feng Jiang
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  2. Jian-ming Yu
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Correspondence to Guang-feng Jiang.

Additional information

This work was partially supported by the National Natural Science Foundation of China (Grant No. 10671009)

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Jiang, Gf., Yu, Jm. Reducibility of hyperplane arrangements. SCI CHINA SER A 50, 689–697 (2007). https://doi.org/10.1007/s11425-007-2075-z

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  • DOI: https://doi.org/10.1007/s11425-007-2075-z

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