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[NA] is Cohen–Macaulay if and only if the rank of the GKZ hypergeometric system HA(β) equals the normalized volume of conv(A) for all complex parameters β ∈ Cd (Saito, 2002). Our refinement here shows that HA(β) has rank strictly larger than the volume of conv(A) if and only if β lies in the Zariski closure (in Cd) of all Zdgraded degrees where the local cohomology ⊕ i<d H i m(C[NA]) is nonzero. We conjecture that the same statement holds even when conv(A) is not a simplex.