We analyze the behavior of the holonomic rank in families of holonomic systems over complex algebraic varieties by providing homological criteria for rank-jumps in this general setting. Then we investigate rank-jump behavior for hypergeometric systems  H A ( β ) H_A(\beta ) arising from a d × n d \times n integer matrix  A A and a parameter β ∈ C d \beta \in \mathbb {C}^d . To do so we introduce an Euler–Koszul functor for hypergeometric families over  C d \mathbb {C}^d , whose homology generalizes the notion of a hypergeometric system, and we prove a homology isomorphism with our general homological construction above. We show that a parameter β ∈ C d \beta \in \mathbb {C}^d is rank-jumping for H A ( β ) H_A(\beta ) if and only if β \beta lies in the Zariski closure of the set of C d \mathbb {C}^d -graded degrees  α \alpha where the local cohomology ⨁ i > d H m i ( C [ N A ] ) α \bigoplus _{i > d} H^i_\mathfrak m(\mathbb {C}[\mathbb {N} A])_\alpha of the semigroup ring C [ N A ] \mathbb {C}[\mathbb {N} A] supported at its maximal graded ideal  m \mathfrak m is nonzero. Consequently, H A ( β ) H_A(\beta ) has no rank-jumps over  C d \mathbb {C}^d if and only if C [ N A ] \mathbb {C}[\mathbb {N} A] is Cohen–Macaulay of dimension  d d .

Abstract. We study the irregularity sheaves attached to the A-hypergeometric D-module M A (β) introduced by I.M. Gel'fand et al. [GGZ87,GZK89], where A ∈ Z d×n is pointed of full rank and β ∈ C d . More precisely, we investigate the slopes of this module along coordinate subspaces.In the process we describe the associated graded ring to a positive semigroup ring for a filtration defined by an arbitrary weight vector L on torusequivariant generators. To this end we introduce the (A, L)-umbrella, a cell complex determined by A and L, and identify its facets with the components of the associated graded ring.We then establish a correspondence between the full (A, L)-umbrella and the components of the L-characteristic variety of M A (β). We compute in combinatorial terms the multiplicities of these components in the L-characteristic cycle of the associated Euler-Koszul complex, identifying them with certain intersection multiplicities.We deduce from this that slopes of M A (β) are combinatorial, independent of β, and in one-to-one correspondence with jumps of the (A, L)-umbrella.

Abstract. For a germ f on a complex manifold X, we introduce a complex derived from the Liouville form acting on logarithmic differential forms, and give an exactness criterion. We use this Liouville complex to connect properties of the D-module generated by f s to homological data of the Jacobian ideal; specifically we show that for a large class of germs the annihilator of f s is generated by derivations. Through local cohomology, we connect the cohomology of the Milnor fiber to the Jacobian module via logarithmic differentials.In particular, we consider (not necessarily reduced) hyperplane arrangements: we prove a conjecture of Terao on the annihilator of 1/f ; we confirm in many cases a corresponding conjecture on the annihilator of f s but we disprove it in general; we show that the Bernstein-Sato polynomial of an arrangement is not determined by its intersection lattice; we prove that arrangements for which the annihilator of f s is generated by derivations fulfill the Strong Monodromy Conjecture, and that this includes as very special cases all arrangements of Coxeter and of crystallographic type, and all multi-arrangements in dimension 3.

Let Q ∈ C[x 1 , . . . , x n ] be a homogeneous polynomial of degree k > 0. We establish a connection between the Bernstein-Sato polynomial b Q (s) and the degrees of the generators for the top cohomology of the associated Milnor fiber. In particular, the integerThe link is provided by the relative de Rham complex and D-module algorithms for computing integration functors.As an application we determine the Bernstein-Sato polynomial b Q (s) of a generic central arrangement Q = k i=1 H i of hyperplanes. In turn, we obtain information about the cohomology of the Milnor fiber of such arrangements related to results of Orlik and Randell who investigated the monodromy.We also introduce certain subschemes of the arrangement determined by the roots of b Q (s). They appear to correspond to iterated singular loci.

In this paper we present algorithms that compute certain local cohomology modules associated to a ring of polynomials containing the rational numbers. In particular we are able to compute the local cohomological dimension of algebraic varieties in characteristic zero. Our approach is based on the theory of Dmodules.