Irregularity of hypergeometric systems via slopes along coordinate   subspaces
  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. 
Mathias Schulze and Uli Walther