Irregularity of hypergeometric systems via slopes along coordinate subspaces

Abstract

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.

44 pages, 3 figures, choose PS or PDF to see figures, new Lemma 2.8 fills gap in previous version of Lemma 2.12, error in previous version of Theorem 3.2 repaired by considering L-holonomic modules in Sections 3.2 and 4.2

References in corpus (1)

• Irregularity of hypergeometric systems via slopes along coordinate subspaces

Cited by in corpus (5)

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• A geometric degree formula for $A$-discriminants and Euler obstructions of toric varieties