Modular Equalities for Complex Reflection Arrangements

Abstract

We compute the combinatorial Aomoto-Betti numbers β p ÔAÕ of a complex reflection arrangement. When A has rank at least 3, we find that β p ÔAÕ 2, for all primes p. Moreover, β p ÔAÕ 0 if p 3, and β 2 ÔAÕ 0 if and only if A is the Hesse arrangement. We deduce that the multiplicity e d ÔAÕ of an order d eigenvalue of the monodromy action on the first rational homology of the Milnor fiber is equal to the corresponding Aomoto-Betti number, when d is prime. We give a uniform combinatorial characterization of the property e d ÔAÕ 0, for 2 d 4. We completely describe the monodromy action for full monomial arrangements of rank 3 and 4. We relate e d ÔAÕ and β p ÔAÕ to multinets, on an arbitrary arrangement.

On the Milnor Monodromy of the Exceptional Reflection Arrangement of Type $G_{31}$

“…It follows that h 1 can have only eigenvalues of order 1, 2, 3 and 6, since any eigenvalue of h 1 has to be a root of at least one of the local Alexander polynomials associated to the singularities of V (G 31 ), see [19], [7,Corollary 6.3.29]. The eigenvalues of order 2 and 3 are excluded by the results in [22]. To prove Theorem 1.1, it remains to exclude the eigenvalues of order 6, and to do this, we apply Theorem 2.1, for d = 60 and k = 10 or k = 50.…”

“…(iii) One can avoid using the results in [22] and exclude the eigenvalues of order 2 and 3 by the same approach as we use in the case of eigenvalues of order 6. These computations correspond to k ∈ {20, 30, 40}.…”

“…is the object of the papers [22] and [9], while in the special case of real reflection groups G, there are some additional results on the higher degree monodromy operators h j (G) :…”

On the minimal value of global Tjurina numbers for line arrangements

“…Note that the arrangements of d lines having points of multiplicity ≥ d − 1 or such that d < 5 are easy to describe, and their monodromy is well-known. In fact, for most line arrangements C, the Alexander polynomial ∆ 1 C (t) is trivial, namely it satisfies (1.5) ∆ 1 C (t) = (t − 1) r−1 , where r is the number of irreducible components of C, see for instance [5,27,39]. Even when the Alexander polynomial ∆ 1 C (t) is non-trivial, possible roots of ∆ 1 C (t) seem to be very restricted.…”