Irregularity of hypergeometric systems via slopes along coordinate subspaces

By Mathias Schulze, Uli Walther. 2006

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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.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7f550e32b4cc260d428295333831204389a5262b

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In Computational Theory and Mathematics

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