Combinatorics of rank jumps in simplicial hypergeometric systems

Abstract

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