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15 April 2008 Irregularity of hypergeometric systems via slopes along coordinate subspaces
Mathias Schulze, Uli Walther
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Duke Math. J. 142(3): 465-509 (15 April 2008). DOI: 10.1215/00127094-2008-011

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 AZd×n is pointed of full rank and βCd. 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

Citation

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Mathias Schulze. Uli Walther. "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

Information

Published: 15 April 2008
First available in Project Euclid: 23 April 2008

zbMATH: 1144.13012
MathSciNet: MR2412045
Digital Object Identifier: 10.1215/00127094-2008-011

Subjects:
Primary: 13N10
Secondary: 14M25 , 16W70

Rights: Copyright © 2008 Duke University Press

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Vol.142 • No. 3 • 15 April 2008
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