Irregularity of hypergeometric systems via slopes along coordinate subspaces

Abstract

We study the irregularity sheaves attached to the A-hypergeometric D-module MA(β) introduced by I. M. Gel'fand and others in [GGZ], [GZK], where A∈Zd×n is pointed of full rank and β∈Cd. 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 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 B. Sturmfels and gives a converse of a theorem of R. Hotta [Ho, Chap. II, §6.2, Th.]: MA(β) is regular if and only if A defines a projective variety