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) arising from a d x n integer matrix A and a parameter β\in \CC^d. To do so we introduce an Euler-Koszul functor for hypergeometric families over \CC^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 βis rank-jumping for H_A(β) if and only if βlies in the Zariski closure of the set of \ZZ^d-graded degrees αwhere the local cohomology \bigoplus_{i<d}H^i_\frakm(\CC[\NN A])_αof the semigroup ring \CC[\NN A] supported at its maximal graded ideal \frakm is nonzero. Consequently, H_A(β) has no rank-jumps over \CC^d if and only if \CC[\NN A] is Cohen-Macaulay of dimension d.