Irregularity of hypergeometric systems via slopes along coordinate subspaces

Abstract

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.

On the parametric behavior of $A$-hypergeometric series

“…When A is homogeneous, this follows from the fact that D/H A (β) is a regular holonomic D-module [Hot91], by the method of canonical series [SST00, Sections 2.5 and 2.6]. If A is not homogeneous, then D/H A (β) has irregular singularities [SW08] (these systems are also known as confluent), but if the domain of expansion is adequately chosen, one can still write A-hypergeometric functions as Nilsson series [DMM12].…”

“…In general, the GG systems corresponding to the hypergeometric equations studied here give rise to the solutions of those systems for generic parameters, as is stated in [GG99, The algorithm to compute canonical series solutions presented in [SST00, Section 2.6] applies to all regular holonomic left D-ideals. For an A-hypergeometric system, regular holonomicity is equivalent to the matrix A being homogeneous [Hot91,SW08], an assumption which we impose for the remainder of this section. …”

Regular b-functions of D-modules

“…That is the case with Examples 1-5 above for generic β. Note also that Schulze and Walther [12] described the slopes of H A (β) along coordinate subvarieties in terms of what they call (A, L)-umbrellas under the condition that the column vectors of A are contained in a proper convex cone with vertex at the origin.…”

Systems of parameters and holonomicity of A-hypergeometric systems

“…Adolphson's proof relies on careful algebraic analysis of the coordinate rings of a collection of varieties whose union is the characteristic variety of the system. Another proof of the holonomicity of an A-hypergeometric system, by Schulze and Walther [SW08], yields a more general result: for a weight vector L from a large family of possibilities, the L-characteristic variety for the L-filtration is a union of conormal varieties and hence has dimension n; holonomicity follows when L = (0, . .…”