Homological Methods for Hypergeometric Families

Details Homological Methods for Hypergeometric Families Homological Methods for Hypergeometric Families

2004-06-18

arxiv.org/abs/math/0406383v2

Mathematics

Algebraic Geometry

Scientific paper

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(\beta) arising from a d x n integer matrix A and a parameter \beta \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 \beta is rank-jumping for H_A(\beta) if and only if \beta lies in the Zariski closure of the set of \ZZ^d-graded degrees \alpha where the local cohomology \bigoplus_{i

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