Mathematics – Algebraic Geometry
Scientific paper
2008-07-02
Mathematics
Algebraic Geometry
32 pages. To appear in Compositio Mathematica. Revisions have been made to the exposition, and the notation has been simplifie
Scientific paper
The holonomic rank of the A-hypergeometric system M_A(\beta) 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(\beta). As consequences, we obtain a refinement of the stratification of the exceptional arrangement by the rank of M_A(\beta) 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(\beta).
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