Irregularity of Hypergeometric Systems via Slopes along Coordinate Subspaces

We study the irregularity sheaves attached to the A-hypergeometric D-module MA(β) introduced by Gel’fand et al., where A ∈ Z d×n is pointed of full rank and β ∈ C. 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 torus equivariant generators. To this end we introduce the (A,L)-umbrella, a simplicial 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 MA(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 MA(β) are combinatorial, independent of β, and in one-to-one correspondence with jumps of the (A,L)-umbrella. This confirms a conjecture of Sturmfels and gives a converse of a theorem of Hotta: MA(β) is regular if and only if A defines a projective variety. 1991 Mathematics Subject Classification. 13N10,14M25,16W70.