Combinatorics of rank jumps in simplicial hypergeometric systems

Let A be an integer d × n matrix, and assume that the convex hull conv(A) of its columns is a simplex of dimension d - 1 not containing the origin. It is known that the semigroup ring ℂ[Ndbl;A] is Cohen-Macaulay if and only if the rank of the GKZ hypergeometric system H A(β) equals the normalized volume of conv(A) for all complex parameters β ε ℂ d (Saito, 2002). Our refinement here shows that H A(β) has rank strictly larger than the volume of conv(A) if and only if β lies in the Zariski closure (in ℂ d) of all Zdbl; d-graded degrees where the local cohomology ⊕ i

Duke Scholars

Author Ezra Miller Mathematics