TY - JOUR

T1 - The characteristic polynomial of a multiarrangement

AU - Abe, Takuro

AU - Terao, Hiroaki

AU - Wakefield, Max

N1 - Funding Information:
* Corresponding author. E-mail addresses: abetaku@math.sci.hokudai.ac.jp (T. Abe), terao@math.sci.hokudai.ac.jp (H. Terao), wakefield@math.sci.hokudai.ac.jp (M. Wakefield). 1 The author is supported by 21st Century COE Program “Mathematics of Nonlinear Structures via Singularities” Hokkaido University. 2 The author has been supported in part by Japan Society for the Promotion of Science. 3 The author has been supported by NSF grant #0600893 and the NSF Japan program.

PY - 2007/11/10

Y1 - 2007/11/10

N2 - Given a multiarrangement of hyperplanes we define a series by sums of the Hilbert series of the derivation modules of the multiarrangement. This series turns out to be a polynomial. Using this polynomial we define the characteristic polynomial of a multiarrangement which generalizes the characteristic polynomial of an arrangement. The characteristic polynomial of an arrangement is a combinatorial invariant, but this generalized characteristic polynomial is not. However, when the multiarrangement is free, we are able to prove the factorization theorem for the characteristic polynomial. The main result is a formula that relates 'global' data to 'local' data of a multiarrangement given by the coefficients of the respective characteristic polynomials. This result gives a new necessary condition for a multiarrangement to be free. Consequently it provides a simple method to show that a given multiarrangement is not free.

AB - Given a multiarrangement of hyperplanes we define a series by sums of the Hilbert series of the derivation modules of the multiarrangement. This series turns out to be a polynomial. Using this polynomial we define the characteristic polynomial of a multiarrangement which generalizes the characteristic polynomial of an arrangement. The characteristic polynomial of an arrangement is a combinatorial invariant, but this generalized characteristic polynomial is not. However, when the multiarrangement is free, we are able to prove the factorization theorem for the characteristic polynomial. The main result is a formula that relates 'global' data to 'local' data of a multiarrangement given by the coefficients of the respective characteristic polynomials. This result gives a new necessary condition for a multiarrangement to be free. Consequently it provides a simple method to show that a given multiarrangement is not free.

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U2 - 10.1016/j.aim.2007.04.019

DO - 10.1016/j.aim.2007.04.019

M3 - Article

AN - SCOPUS:34547653205

VL - 215

SP - 825

EP - 838

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 2

ER -