Homological methods for hypergeometric families Academic Article

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 ( ) H_A(\beta ) arising from a d n d imes n integer matrix A A and a parameter C d \beta in mathbb {C}^d . To do so we introduce an EulerKoszul 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 C d \beta in mathbb {C}^d is rank-jumping for H A ( ) H_A(\beta ) if and only if \beta lies in the Zariski closure of the set of C d mathbb {C}^d -graded degrees alpha where the local cohomology i > d H m i