Combinatorics of rank jumps in simplicial hypergeometric systems

Let $A$ be an integer $d \times n$ matrix, and assume that the convex hull $\operatorname{conv}(A)$ of its columns is a simplex of dimension $d-1$ not containing the origin. It is known that the semigroup ring $\mathbb{C}[\mathbb{N} A]$ is Cohen-Macaulay if and only if the rank of the GKZ hypergeometric system $H_A(\beta)$ equals the normalized volume of $\operatorname{conv}(A)$ for all complex parameters $\beta \in \mathbb{C}^d$ (Saito, 2002). Our refinement here shows that $H_A(\beta)$ has rank strictly larger than the volume of $\operatorname{conv}(A)$ if and only if $\beta$ lies in the Zariski closure (in  $\mathbb{C}^d$) of all $\mathbb{Z}^d$-graded degrees where the local cohomology $\bigoplus_{i < d} H^i_\mathfrak{m}(\mathbb{C} [\mathbb{N}A])$ is nonzero. We conjecture that the same statement holds even when $\operatorname{conv}(A)$ is not a simplex.