Homological Methods for Hypergeometric Families

We analyze the behavior of the holonomic rank in families of holonomicsystems over complex algebraic varieties by providing homological criteria for rank-jumpsin this general setting. Then we investigate rank-jump behavior for hypergeometric sys-tems HA(β) arising from a d × n integer matrix A and a parameter β ∈ C. To doso we introduce an Euler–Koszul functor for hypergeometric families over C, whose ho-mology generalizes the notion of a hypergeometric system, and we prove a homologyisomorphism with our general homological construction above. We show that a parameterβ ∈ C is rank-jumping for HA(β) if and only if β lies in the Zariski closure of the set ofZ-graded degrees α where the local cohomologyL iHm(C[NA])α of the semigroupring C[NA] supported at its maximal graded ideal m is nonzero. Consequently, HA(β)has no rank-jumps over C if and only if C[NA] is Cohen–Macaulay of dimension d. DEPARTMENT OF MATHEMATICS, HARVARD UNIVERSITY, CAMBRIDGE, MACurrent address: Department of Mathematics, University of Pennsylvania, Philadelphia, PAE-mail address: [email protected] SCHOOL OF MATHEMATICS, UNIVERSITY OF MINNESOTA, MINNEAPOLIS, MNE-mail address: [email protected] DEPARTMENT OF MATHEMATICS, PURDUE UNIVERSITY, WEST LAFAYETTE, INE-mail address: [email protected]