Homological methods for hypergeometric families
Laura Felicia Matusevich, Ezra Miller, Uli Walther · Journal of the American Mathematical Society · 2005
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