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Restrictions of Free Arrangements and the Division Theorem

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Part of the book series: Springer INdAM Series ((SINDAMS,volume 19))

Abstract

This is a survey and research note on the modified Orlik conjecture derived from the division theorem introduced in Abe (Invent. Math. 204(1), 317–346, 2016). The division theorem is a generalization of classical addition-deletion theorems for free arrangements. The division theorem can be regarded as a modified converse of the Orlik’s conjecture with a combinatorial condition, i.e., an arrangement is free if the restriction is free and the characteristic polynomial of the restriction divides that of an arrangement. In this article we recall, summarize, pose and re-formulate some of results and problems related to the division theorem based on Abe (Invent. Math. 204(1), 317–346, 2016), and study the modified Orlik’s conjecture with partial answers.

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Acknowledgements

The author is grateful to the referee for the careful reading of this paper with a lot of important comments. This work is partially supported by JSPS Grants-in-Aid for Young Scientists (B) No. 24740012.

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Authors and Affiliations

  1. Institute of Mathematics for Industry, Kyushu University, 819-0395, Fukuoka, Japan

    Takuro Abe

Authors
  1. Takuro Abe
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Correspondence to Takuro Abe .

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica, Università di Pisa, Pisa, Italy

    Filippo Callegaro

  2. Dipartimento di Matematica, University of Padova, Padova, Italy

    Giovanna Carnovale

  3. Dipartimento di Matematica, University of Bologna, Bologna, Italy

    Fabrizio Caselli

  4. Dipartimento di Matematica, Sapienza University of Rome, Roma, Italy

    Corrado De Concini

  5. Dipartimento di Matematica, Sapienza University of Rome, Roma, Italy

    Alberto De Sole

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Abe, T. (2017). Restrictions of Free Arrangements and the Division Theorem. In: Callegaro, F., Carnovale, G., Caselli, F., De Concini, C., De Sole, A. (eds) Perspectives in Lie Theory. Springer INdAM Series, vol 19. Springer, Cham. https://doi.org/10.1007/978-3-319-58971-8_14

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