Homological Methods for Hypergeometric Families
Loading...
Date
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
Description
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.