  We study the irregularity sheaves attached to the $A$-hypergeometric
$D$-module $M_A(\beta)$ introduced by Gel'fand et al., 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
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 $M_A(\beta)$. 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 Sturmfels and gives a converse
of a theorem of Hotta: $M_A(\beta)$ is regular if and only if $A$ defines a
projective variety.

Algebraic Geometry;Combinatorics