We study the irregularity sheaves attached to the -hypergeometric D-module introduced by I. M. Gel'fand and others in [GGZ], [GZK], where is pointed of full rank and . 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 on torus-equivariant generators. To this end, we introduce the -umbrella, a cell complex determined by and , and identify its facets with the components of the associated graded ring.
We then establish a correspondence between the full -umbrella and the components of the -characteristic variety of . We compute in combinatorial terms the multiplicities of these components in the -characteristic cycle of the associated Euler-Koszul complex, identifying them with certain intersection multiplicities.
We deduce from this that slopes of are combinatorial, independent of , and in one-to-one correspondence with jumps of the -umbrella. This confirms a conjecture of B. Sturmfels and gives a converse of a theorem of R. Hotta [Ho, Chap. II, §6.2, Th.]: is regular if and only if defines a projective variety
"Irregularity of hypergeometric systems via slopes along coordinate subspaces." Duke Math. J. 142 (3) 465 - 509, 15 April 2008. https://doi.org/10.1215/00127094-2008-011