The rank of a hypergeometric system

Abstract

The holonomic rank of the A-hypergeometric system M_A(β) is the degree of the toric ideal I_A for generic parameters; in general, this is only a lower bound. To the semigroup ring of A we attach the ranking arrangement and use this algebraic invariant and the exceptional arrangement of nongeneric parameters to construct a combinatorial formula for the rank jump of M_A(β). As consequences, we obtain a refinement of the stratification of the exceptional arrangement by the rank of M_A(β) and show that the Zariski closure of each of its strata is a union of translates of linear subspaces of the parameter space. These results hold for generalized A-hypergeometric systems as well, where the semigroup ring of A is replaced by a nontrivial weakly toric module M contained in \CC[\ZZ A]. We also provide a direct proof of the result of M. Saito and W. Traves regarding the isomorphism classes of M_A(β).

32 pages. To appear in Compositio Mathematica. Revisions have been made to the exposition, and the notation has been simplified

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• GKZ-Generalized Hypergeometric Systems in Mirror Symmetry of Calabi-Yau Hypersurfaces
• Generalized Hypergeometric Functions and Rational Curves on Calabi-Yau Complete Intersections in Toric Varieties
• Equivariant D-modules
• Hypergeometric D-modules and twisted Gauss-Manin systems
• Binomial D-modules
• Generating functions for intersection numbers on moduli spaces of curves
• Isomorphism classes of A-hypergeometric systems
• The A-hypergeometric System Associated with a Monomial Curve

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• Algebraic aspects of hypergeometric differential equations
• Gevrey expansions of hypergeometric integrals I
• Virtual Resolutions of Monomial Ideals on Toric Varieties
• Characteristic cycles and Gevrey series solutions of $A$-hypergeometric systems
• Gevrey solutions of the irregular hypergeometric system associated with an affine monomial curve
• A Zariski theorem for monodromy of $A$-hypergeometric systems
• On the rank of an $A$-hypergeometric $D$-module versus the normalized volume of $A$
• Irregular hypergeometric D-modules
• On the parametric behavior of A-hypergeometric series
• A-graded methods for monomial ideals
• Hypergeometric Functions for Projective Toric Curves