The characteristic polynomial of a multiarrangement

Takuro Abe, Hiroaki Terao, Max Wakefield

Research output: Contribution to journalArticlepeer-review

23 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)825-838
Number of pages14
JournalAdvances in Mathematics
Volume215
Issue number2
DOIs
Publication statusPublished - Nov 10 2007
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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