On the Jacobian ideal of central arrangements

Let A denote a central hyperplane arrangement of rank n in affine space K over an infinite field K and let l1, . . . , lm ∈ R := K[x1, . . . , xn] denote the linear forms defining the corresponding hyperplanes, along with the corresponding defining polynomial f := l1 · · · lm ∈ R. Let Jf denote the ideal generated by the partial derivatives of f and let I designate the ideal generated by the (m−1)-fold products of l1, . . . , lm. This paper is centered on the relationship between the two ideals Jf , I ⊂ R, their properties and two conjectures related to them. Some parallel results are obtained in the case of forms of higher degrees provided they fulfill a certain transversality requirement. INTRODUCTION Let A denote a central hyperplane arrangement of rank n in affine space Kn over an infinite field K and let l1, . . . , lm ∈ R := K[x1, . . . , xn] denote the linear forms defining the corresponding hyperplanes. Set f := l1 · · · lm ∈ R, the defining polynomial of A. The module of logarithmic derivations associated to f is defined as Derlog(A) := {θ ∈ Der(R)|θ(f) ⊂ 〈f〉}. Its R-module structure is well known if char(K) does not divide m = deg(f): Derlog(A) = Syz(Jf )⊕RθE. Here Syz(Jf ) ⊂ R n is up to the identification Der(R) = Rn the syzygy module of the partial derivatives of the f and θE = ∑n i=1 xi ∂ ∂xi is the Euler derivation. In the case where the arrangement is generic, – meaning that every subset of {l1, . . . , lm} with n elements is K-linearly independent – and that m ≥ n + 1, Rose-Terao ([9]) and Yuzvinsky ([16]) have established that the homological dimension of Derlog(A) is n−2. The statement is equivalent to having depth(R/Jf ) = 0, or still that the irrelevant maximal ideal m := 〈x1, . . . , xn〉 is an associated prime of R/Jf . In their beautiful paper, Rose and Terao took the approach of establishing explicit free resolutions of the modules Ωq(A) of logarithmic q-forms, for 0 ≤ q ≤ n − 1, in the spirit of Lebelt’s work ([5]). Then, drawing on the isomorphism Derlog(A) ≃ Ωn−1(A), they derived the free resolution of the module of logarithmic derivations, and hence of R/Jf as well. Motivated by this, the present authors consider the problem under a more intrinsic perspective, trying to envisage what can be said, avoiding details of free resolutions, by drawing upon a few well established facts coming from commutative algebra. As such, it seemed natural to ask if there is any result of this sort in the case of forms f1, . . . , fm ∈ R,m ≥ 2, of degrees deg(fi) = di ≥ 2, i = 1, . . . ,m. Accordingly, the paper is focused on two major aspects: first, the nature of Jf from the ideal theoretic and homological point of view, for a central generic hyperplane arrangement; second, the nature of Jf for arrangements of forms that are isolated singularities. 2010 Mathematics Subject Classification. Primary 13A30; Secondary 13C15, 14N20, 52C35.