We study the irregularity sheaves attached to the $A$ -hypergeometric D-module $M_A(\beta)$ introduced by I. M. Gel'fand and others in [GGZ], [GZK], where $A\in\mathbb{Z}^{d\times n}$ is pointed of full rank and $\beta\in\mathbb{C}^d$ . 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 torus-equivariant 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 $M_A(\beta)$ . 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 $M_A(\beta)$ are combinatorial, independent of $\beta$ , and in one-to-one correspondence with jumps of the $(A,L)$ -umbrella. This confirms a conjecture of B. Sturmfels and gives a converse of a theorem of R. Hotta [Ho, Chap. II, §6.2, Th.]: $M_A(\beta)$ is regular if and only if $A$ defines a projective variety