Homological methods for hypergeometric families

Abstract

We analyze the behavior of the holonomic rank in families of holonomic systems over complex algebraic varieties by providing homological criteria for rank-jumps in this general setting. Then we investigate rank-jump behavior for hypergeometric systems  H A ( β ) H_A(\beta ) arising from a d × n d \times n integer matrix  A A and a parameter β ∈ C d \beta \in \mathbb {C}^d . To do so we introduce an Euler–Koszul functor for hypergeometric families over  C d \mathbb {C}^d , whose homology generalizes the notion of a hypergeometric system, and we prove a homology isomorphism with our general homological construction above. We show that a parameter β ∈ C d \beta \in \mathbb {C}^d is rank-jumping for H A ( β ) H_A(\beta ) if and only if β \beta lies in the Zariski closure of the set of C d \mathbb {C}^d -graded degrees  α \alpha where the local cohomology ⨁ i > d H m i ( C [ N A ] ) α \bigoplus _{i > d} H^i_\mathfrak m(\mathbb {C}[\mathbb {N} A])_\alpha of the semigroup ring C [ N A ] \mathbb {C}[\mathbb {N} A] supported at its maximal graded ideal  m \mathfrak m is nonzero. Consequently, H A ( β ) H_A(\beta ) has no rank-jumps over  C d \mathbb {C}^d if and only if C [ N A ] \mathbb {C}[\mathbb {N} A] is Cohen–Macaulay of dimension  d d .