THE CHARACTERISTIC POLYNOMIAL OF A MULTIARRANGEMENT
Preprint Series # 818 ABE, TAKURO and TERAO, HIROAKI and WAKEFIELD, MAX THE CHARACTERISTIC POLYNOMIAL OF A MULTIARRANGEMENT. (2006); AbstractGiven 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 arragnement. 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.
